But does that not negate the need to embrace Jesus as Savior and Lord?

Who ever said that it was not necessary to accept Jesus as Savior for cancelling the guilt and remembrance of past and present sins? Did I ever say that? I do not recall ever stating that.

But doing good works is a human effort.

They are, huh? All by onesself?

Wrong!

The freewill desire, the capacity, and the resources to do good works are from the Lord:

Philippians 2:13 for God is at work in you, both to will and to work for His good pleasure.

James 1:17 Every good endowment and every perfect gift is from above, coming down from the Father of Lights with whom there is no variation or shadow of alteration.

But was not the rant and diatribe of St Paul in the New-Testament books of Romans and Galatians expressly condemnatory of trying to be self-justified by attempting to do good works?

Paul neither ranted nor diatribed, and his dissertations concerning (and not necessarily only "regarding") faith and grace apart from works had absolutely nothing to do with the mandatory requirement of Jesus that His followers: "Be perfect, as their Heavenly Father is perfect" - after chastising them with the criticism:

Luke 6:46 "Why do you call me 'Lord, Lord,' and not do what I tell you?

Paul's essays in both Romans and Galatians had everything to do with reminding the saints that their sins could not be absolved and atoned for by they themselves attempting to do hit-and-miss good works whenever they sometimes did do them.

Nor could the sins of their occasional faithlessness infrequently doubting Him be self-absolved, as they often disbelieved in non-faith that He not necessarily could not, but would not, provide for them - soon enough or easy enough - without they themselves going ahead and doing impulsive or impetuous sin because of faithless presumption.

Why do preachers blatter to congregants, who they do not want to be persecuted by for exposing and reprimanding them for certain obvious sins which such congregants perpetuate, that: "Oh, good works are not needed for salvation. I'm OK, you're OK. Jesus saved us, because we could not do it ourselves. Jesus, not us, paid the price. Jesus did it all for us. Our perfection is in Him, not ourselves. Once saved, always saved. We have already been sanctified and perfected for all time, like it says in Hebrews chapter 10. All we have to do now is accidentally sin without the Law in living-in-the-Spirit freedom being under non-solicited-duress temptation, and - albeit reluctantly - be at liberty to live like the Devil, because our past sins and any present ones or future ones we did, do, or will do have already been covered by the blood of the Lamb?"

But, is not the most simple and purest definition of "the gospel" being that all it takes to be saved is to believe in Jesus?

Not so:

James 2:19 You believe that God is one. OK, even the demons believe...and shudder.

Ezekiel 33:31 And they come to you as people come, and they sit before you as my people, and they hear what you say but they will not do it; for with their lips they show much love, but their heart is set on their gain.
Matthew 15:8 This people honors me with their lips, but their heart is far from me....
Mark 7:6 And He said to them: "Well did Isaiah prophesy of you hypocrites, as it is written: "This people honors me with their lips, but their heart is far from me..."

Matthew 21:31 Which of the two [actually] did the will of his father?" They said, "The first." Jesus said to them: "Truthfully, I say to you, the tax collectors and the harlots go into the kingdom of God instead of you."

Just believe in, praise, and worship God, but plan to perhaps sin after church, because it does not really matter, being that no one is justified by works, OK?

Not OK:

Galatians 2:16 ...a human is not justified by works of the law but through faith in Jesus Christ, even we have believed in Christ Jesus, in order to be justified by faith in Christ, and not by works of the Law, because by works of the Law shall no one be justified. [And if the reader has trouble with that relating to good-works requirement, re-read the explanation above about Law, trying to perfectly obey the Law, somewhat obeying the Law -- all of which never ever absolves past sins....which absolution, atonement, and redemption only the crucified-and-risen Christ provided and provides]

Remember the Black Monolith in the movie 2001: A Space Odessey. That was the Law, and every time humankind discovered it, or it was presented to them, and they tried to reach for the Law, they got smarter, and progressed.

James 2:21 Was not Abraham our father justified by works, when he [pseudo-murderously] offered [or was in the process of completely and lethally offering] his son Isaac upon the altar?
James 2:24 You see that a human is justified by [good, not evil?] works and not by faith [in what?] alone.
James 2:25 And in the same way was not also Rahab the harlot justified by works when she received the messengers and sent them out another way?

Words, words, words. A mindless rote litany of mere words. A save-onesself boring mantra of praise songs without morality cleanup?

Hosea 4:13 "They sacrifice on the tops of the mountains, and make offerings upon the hills, under oak, poplar, and terebinth, because their shade is good. Therefore your daughters play the harlot, and your brides commit adultery.
4:14 I will not punish your daughters when they play the harlot, nor your brides when they commit adultery; for the men themselves go aside with harlots, and sacrifice with cult prostitutes, and a people without understanding shall come to ruin.
6:6 For I desire steadfast love and not sacrifice, the knowledge of God, rather than burnt offerings.
8:13 They love sacrifice; they sacrifice flesh and eat it; but the LORD has no delight in them. Now He will remember their iniquity, and punish their sins; they shall return to Egypt.
12:11 If there is iniquity in Gilead they shall surely come to nothing; if in Gilgal they sacrifice bulls, their altars also shall be like stone heaps on the furrows of the field.
13:2 And now they sin more and more, and make for themselves molten images, idols skillfully made of their silver, all of them the work of craftsmen. "Sacrifice to these," they say. Men kiss calves!"

Amos 4:5 "...offer a sacrifice of thanksgiving of that which is leavened, and proclaim freewill offerings, publish them; for so you love to do, people of Israel!" says the Lord GOD.

God accepts us as we are, and all we have to do is believe in Jesus by faith?

Not so:

Matthew 22:11 But when the king came in to look at the guests, he saw there a man who had no wedding garment;
Matthew 22:12 and he said to him: "Friend, how did you get in here without a wedding garment?" And he was speechless.
Matthew 22:13 Then the king said to the attendants: "Bind him hand and foot, and throw him into the Outer Darkness; there people will weep and gnash their teeth."
Matthew 22:14 For many are called, but few are chosen.

But aren't we dignified with self-esteem, honorable, rich, see clearly, and clothed in His righteousness alone?

Revelation 3:17 For you say: "I am rich, I have prospered, and I need nothing" -- not knowing that you are wretched, pitiable, poor, blind, and naked.

Isaiah 47:1 Come down and sit in the dust, virgin daughter of Babylon; sit on the ground without a throne, daughter of the Chaldeans! For you shall no more be called tender and delicate.
47:2 Take the millstones and grind meal, put off your veil, strip off your robe, uncover your legs, pass through the rivers.
47:3 Your nakedness shall be uncovered, and your shame shall be seen. I will take vengeance, and I will spare no human.
47:4 Our Redeemer, the LORD of hosts is His name: the Holy One of Israel.

We do not redeem ourselves by pretending that we were or are clothed in Jesus when in fact we happen to actually be hideously, shamefully, and immodestly misdressed pornographically. If we repent about being publicly mopheaded, sleevesless, slacksless, socksless...and redress for decency...then [and only then] is our applicable Redeemer the Holy One of Israel.

Simply because we believe in Jesus, believe His gospel, and supposedly are not under Law but under Grace (whatever that means and does not mean), we are no longer bound by the following?

Jeremiah 2:20 "For long ago you broke your yoke and burst your bonds; and you said: "I will not serve." Yes, upon every high hill and under every green tree you bowed down as a harlot.
2:21 Yet I planted you a choice vine, wholly of pure seed. How then have you turned degenerate and become a wild vine?
2:22 Though you wash yourself with lye and use much soap, the stain of your guilt is still before me," says the Lord GOD.
2:23 How can you say: "I am not defiled, I have not gone after the Baals?" Look at your way in the valley; know what you have done: a restive young camel crossing her tracks,
2:24 a wild ass used to the wilderness, in her heat sniffing the wind! Who can restrain her lust? None who seek her need weary themselves; in her month they will find her.
2:25 Keep your feet from going unshod and your throat from thirst. But you said: "It is hopeless, for I have loved strangers, and after them I will go."

Do not "wear short shorts, let your hair down and kick off your shoes" [contrary to some lewd lyrics of country singers]...unless alone with your own spouse, if done for sexually-related purposes.

Summarized by (while not overriding the specifics of the previous):

First Corinthians 12:23 ...and those parts of the body which we think less honorable we invest with the greater honor, and our unpresentable parts are treated with greater modesty,
First Corinthians 12:24 which our more presentable parts do not require. But God has so composed the body, giving the greater honor to the inferior part...

First Timothy 2:8 I desire then that in every place the men should pray, lifting holy hands without anger or quarreling;
2:9 also that women should adorn themselves modestly and decently in appropriate apparel, not with plaited hair or gold or pearls or costly clothing
2:10 but by good works, as befits women who profess religion.
2:11 Have a woman learn in silence with all submissiveness.
2:12 I allow no woman to teach or to have authority over men; she is to keep silent.
2:13 For Adam was formed first, then Eve;
2:14 and Adam was not deceived, but the woman was deceived and became a transgressor.
2:15 Yet woman will be saved by bearing children, if she continues in faith and love and holiness, with sensibility.

Second Corinthians 5:4 For while we are still in this tent, we sigh with anxiety; not that we would be unclothed, but that we would be further clothed, so that what is mortal may be swallowed up by life.

Look at it this way.

Is faith the only work we need to do?

John 6:28 Then they said to Him: "What must we do, to be doing the works of God?"
John 6:29 Jesus answered them: "This is the work of God, that you believe in Him whom He has sent."

But:

James 2:19 You believe that God is one? Fine - even the demons believe, and shudder.

Mark 1:24 ...and he cried out: "What have you to do with us, Jesus of Nazareth? Have you come to destroy us? I know who you are, the Holy One of God."
Luke 4:34 "Ah! What have you to do with us, Jesus of Nazareth? Have you come to destroy us? I know who you are, the Holy One of God."

So, simply "believe in Jesus," and forget the Mosaic Law?

Keep in mind that Jesus did not advocate ignoring or disobeying the Mosaic Law:

Matthew 5:18 Truthfully I say to you, until heaven and Earth pass away, not an iota, not a dot, will pass from the Law until all is accomplished.

Accomplished by who?

Matthew 5:20 For I tell you, unless your [not "Christ's"] righteousness exceeds that of the scribes and Pharisees, you will never enter the kingdom of heaven.

Righteous in accordance with the Law is not neutralized nor trumped by the gospel of forgiveness, but goes beyond it by requiring godly attitude and motivation along with and not merely physical actions of obedience.

Drive on the right side of the road out of love in your heart for fellow motorists, not simply because a cop will arrest you if you do not obey traffic laws.

And men, if you have more than one car, you are not guilty of either bigamy nor polygamy, but tenderly care for any additional vehicles beyond only one if you have the time and can afford more. Follow in the non-condemned-for-having-harems examples of Abraham [Sarah and Hagar], Jacob [Leah, Rachel, Bilhah, Zilpah], Gideon [Judges 8:30 = Now Gideon had seventy sons, his own offspring, for he had many wives], and more:

First Chronicles 4:5 Ashhur, the father of Tekoa, had two wives, Helah and Naarah
First Chronicles 8:8 And Shaharaim had sons in the country of Moab after he had sent away Hushim and Baara his wives.
First Chronicles 14:3 And David took more wives [Abigail, Ahinoam, Maacah, Haggith, Abital, Eglah and First Chronicles 3:9 = All these were David's sons, besides the sons of the concubines...] in Jerusalem, and David begot more sons and daughters.

Second Samuel 15:16 So King David went out, and all his household after him. And the king left [his] ten concubines to keep the house.
Second Samuel 20:3 And David came to his house at Jerusalem; and the king took the ten concubines whom he had left to care for the house [and who his son Absalom had lain with], and put them in a house under guard, and provided for them, but did not go in to them. So they were shut up until the day of their death, living as if in widowhood.

Second Chronicles 9:5 And [the Queen of Sheba] said to [King David]: "The report was true which I heard in my own land of your affairs and of your wisdom,
Second Chronicles 9:7 Happy are your wives! Happy are these your servants, who continually stand before you and hear your wisdom!

As for Solomon, he sinned not because of polygamous overabundance of horses, wives, gold, and silver [see Deuteronomy 17:16-17] but rather mismating with disbelieving pagans [see Second Corinthians 6:14].

Indeed, his God-given wisdom acknowledged:

Ecclesiastes 2:8 I also gathered for myself silver and gold and the treasure of kings and provinces; I got singers, both men and women, and many concubines, man's delight. [We're not talking porn models, masseuse or escort prostitutes, porn pics or vids or sculptures...but instead permanent spouses belonging to him only in marriage]

...which concubine(s) acquisition and retention is not allowed for either bishops or deacons:

First Timothy 3:2 Now a bishop must be above reproach, the husband of one wife, temperate, sensible, dignified, hospitable, an apt teacher...
First Timothy 3:12 Let deacons be the husband of one wife, and let them manage their children and their households well...

Second Chronicles 11:21 [Sheik?] Rehoboam loved Maacah the daughter of Absalom above all his wives and concubines (he took eighteen wives and sixty concubines, and had twenty-eight sons and sixty daughters)
Second Chronicles 13:21 But [Tribal-Chief?] Abijah grew mighty. And he took fourteen wives, and had twenty-two sons and sixteen daughters.

Second Chronicles 24:2 And Joash did what was right in the eyes of the LORD all the days of Jehoiada the priest.
Second Chronicles 24:3 Jehoiada got for him two wives, and he had sons and daughters. [and all the dual credit-card shopping charges, menstrual periods sexlessness at times for one, headache sexlessness at times for another, child-birthing and child-rearing expenses, etc. that accompanied that]

No need to wait for Nirvana to have the 70 virgins!

Is obeying the Law that important?

Matthew 5:29 If your right eye causes you to sin, pluck it out and throw it away; it is better that you lose one of your members than that your whole body be thrown into Hell.
Matthew 5:30 And if your right hand causes you to sin, cut it off and throw it away; it is better that you lose one of your members than that your whole body go into Hell.

That severity of such drastic inhibitions will hopefully not be necessary, but simply take the forms of turning off the TV, cancelling the porno cable TV subscription, going the longer way home or a different route, wearing non-prescription sunglasses if near-sighted, and more.

However, when God allows soldiers wounded to the extent of loss of limbs or eyesight from overseas battles, such is literally true, and a lesser application would be the Lord letting his servants slip or fall so as to get whatever body parts broken or fractured and temporarily immobilized in a sling, splint, or cast.

Romans 8:28 We know that in everything God works for good with those who love Him, who are called according to His purpose.

Without the Law, sin is not defined. If sin is not defined, how can anyone repent for doing what is categorized as sinful?

If there is no Law, and sin is not defined, there is no need for atonement, absolution, nor redemption.

But does wanting to do good works, and then actually doing them, result in boasting?

Sure does, but boasting about what?

Psalm 34:2 My soul makes its boast in the LORD; let the afflicted hear and be glad.

Romans 2:23 You who boast in the Law, do you dishonor God by breaking the Law?
Romans 11:18 ...do not boast over the branches. If you do boast, remember it is not you who support the root, but the root who supports you.

First Corinthians 1:31 ...therefore, as it is written: "Let him who boasts, boast of the Lord."
3:21 So let no one boast of men. For all things are yours...
4:7 For who sees anything different in you? What have you that you did not receive? If then you received it, why do you boast as if it were not a gift?

Second Corinthians 1:12 For our boast is this, the testimony of our conscience that we have behaved in the world, and still more toward you, with holiness and godly sincerity, not by earthly wisdom but by the grace of God.
10:8 For even if I boast a little too much of our authority, which the Lord gave for building you up and not for destroying you, I shall not be put to shame.
10:15 We do not boast beyond limit, in other men's labors; but our hope is that as your faith increases, our field among you may be greatly enlarged
10:17 "Let him who boasts, boast of the Lord."
11:10 As the truth of Christ is in me, this boast of mine shall not be silenced in the regions of Achaia.
11:16 I repeat, let no one think me foolish; but even if you do, accept me as a fool, so that I too may boast a little.
11:18 being that many boast of worldly things, I too will boast.

Jesus saves, not good works done by any human?

Titus 1:16 They profess to know God, but they deny Him by their works; they are detestable, disobedient, unfit for any good work.

Revelation 14:12 Here is a call for the endurance of the saints, those who keep the commandments of God and the faith of Jesus.

Revelation 20:12 And I saw the dead, great and small, standing before the throne, and books were opened. Also another book was opened, which is the book of life. And the dead were judged by what was written in the books, by what they had done.
Revelation 20:13 And the sea gave up the dead in it, Death and Hades gave up the dead in them, and all were judged by what they had done. [A broad inclusion indeed, encompassing both faith in Jesus and good works done with His permission and power].

Revelation 22:14 [KJ21, not RSV] Blessed are they who do His commandments, that they may have the right to the Tree of Life, and may enter in through the gates into The City.

After the heavens and Earth have [post-Millennially] been annihilated in the twinkling-of-an-eye instant with fervent heat with a loud noise [see First Corinthians 15:52 and Second Peter 3:7-12], and both the resurrected wicked dead plus the resurrected dead and raptured righteous saints stand before The Almighty King in Final Judgment...will the saints be exonerated by the fact that they believed the gospel? No. That they accepted Jesus as their personal Savior and Lord? No. That they presumed that they were saved by grace through faith apart from works of Law? No.

Well, what then? By having done good works?

Matthew 25:14 It will be as when a man going on a journey called his servants and entrusted to them his property;
25:15 to one he gave five talents, to another two, to another one, to each according to his ability. Then he went away.
25:16 He who had received the five talents went at once and traded with them [a good work]; and he made five talents more.
25:18 But he who had received the one talent went and dug in the ground and hid his master's money.
25:19 Now after a long time the master of those servants came and settled accounts with them.
25:20 And he who had received the five talents came forward, bringing five talents more, saying: "Master, you delivered to me five talents; here I [having done good works] have made five talents more."
25:21 His master said to him: "Well done, good and faithful servant; you have been faithful over a little, I will set you over much; enter into the joy of your master."
25:24 He also who had received the one talent came forward, saying: "Master, I knew you to be a hard man, reaping where you did not plant, and gathering where you did not winnow;
25:25 so I was afraid, and I went and hid your talent in the ground. Here you have what is yours."
25:26 But his master answered him: "You wicked and slothful servant! You knew that I reap where I have not planted, and gather where I have not winnowed?
25:27 Then you ought to have invested my money with the bankers, and at my coming I should have received what was my own with interest.
25:28 So take the talent from him, and give it to him who has the ten talents.
25:29 For to every one who has will more be given, and he will have abundance; but from him who has not, even what he has will be taken away.
25:30 And throw the worthless servant into The Outer Darkness; there men will weep and gnash their teeth."

Matthew 25:31 "When the Son of Man comes in His glory, and all the angels with Him, then He will sit on His Glorious Throne.
25:32 in front of Him will be gathered all the nations, and He will separate them one from another as a shepherd separates the sheep from the goats,
25:33 and He will place the sheep at his right hand, but the goats at the left.
25:34 Then the King will say to those at His right hand: "Come, blessed of my Father, inherit the kingdom prepared for you from the foundation of the world;
25:35 because you did the good works of giving those of mine food when hungry, drink when thirsty, housing when a stranger, clothing when naked, medical care when sick, and visitation when in jail...
25:41 Then He will say to those at His left hand: "Get outa here, you cursed, into The Eternal Fire prepared for The Devil and his [demons];
25:42 because you did not do the good works the sheep did for others of mine.
25:46 And the wicked [proverbial/figurative/allegorical goats] will go away into eternal punishment, but the righteous [proverbial/symbolic/metaphorical sheep] into eternal life."

How many good works must I do to be saved and get into Paradise?

As many as you can.

We are all human; nobody is perfect. We cannot be perfect and always do only good. You expect the impossible.

Whether or not you can or cannot be totally good 24/7 is irrelevant; you must always be completely good - however and whatever it takes.

Deuteronomy 30:11 "For this commandment which I command you this day is not too hard for you, neither is it far off."

You can do all things through Christ who strengthens you [Philippians 4:13]. With humans it is impossible, but with God, all things are possible. [Matthew 19:26, Mark 9:23 & 10:27]

That's with God, not with us. Even Scripture says I cannot always and only do good works, according to:

Genesis 8:21 And when the LORD smelled the pleasing aroma [of Noah's thank offering], the LORD said in His heart: "I will never again curse the ground because of humanity, for the imagination of the human heart is evil from their youth; neither will I ever again destroy every living creature as I have done.

Joshua 24:19 But Joshua said to the people: "You cannot serve the LORD, because He is a holy God; He is a jealous God; He will not forgive your transgressions or your sins."

Ecclesiastes 7:20 Surely there is not a righteous person on Earth who does good and never sins.

Jeremiah 17:9 The heart is deceitful above all things, and desperately corrupt; who can understand it?

First John 1:8 If we say we have no sin, we deceive ourselves, and the truth is not in us.

Clearly, all of us are then in a hopeless dilemma, which can only be resolved with humble penitence in and with Christ Jesus.

The doing of whatever good works (as the Bible defines what they are and are not) which one has the opportunity to do, along with the honest intention and expressed desire to do nothing but good works perfectly 24/7, is required and mandatory to not merely indicate who is a real Christian, but even to actually be an authentic Christian.

Cowardly or non-proselytizing silent belief in one's heart (about Christ's gospel) is not enough to prove being saved nor be saved, if one has the capacity and the opportunity to communicate their belief (in the atonement against sins provided by Christ Jesus) by mouth, pen, or however.

God in Christ does not tolerate nor accept part-time love, nor any individual comprising His Body, His Bride, the Church, having competing husbands.

Second Corinthians 1:19 For the Son of God, Jesus Christ, whom we preached among you, Silvanus and Timothy and I, was not Yes and No; but in Him it is always Yes.