Why is there a day of judgement?

Jesus isn’t teaching a works based salvation in Matthew 25:31-46, rather it’s about what people have done in this life to or for Christ.

Justification before God, on behalf of what Christ accomplished is by grace through faith in Jesus alone: (Romans 5:1-5, Ephesians 2:8-9)

The fact of what Jesus is getting at is the fruit of their deeds, which shows whether or not they are truly His sheep.

Those who are truly His people will have demonstrated good works towards others because they had been called to do so:

“For by grace you have been saved through faith, and that not of yourselves; it is the gift of God, not of works, lest anyone should boast.” ‭‭Ephesians‬ ‭2:8-9

“For we are his workmanship, having been created in Christ Jesus for good works that God prepared beforehand so we may do them.” ‭‭Ephesians‬ ‭2:10‬

Those who do not belong to Christ, we’re self seeking & did not honor Christ with their deeds, a few examples in scripture are as follows:

“All is pure to those who are pure. But to those who are corrupt and unbelieving, nothing is pure, but both their minds and consciences are corrupted. They profess to know God but with their deeds they deny him, since they are detestable, disobedient, and unfit for any good deed.” ‭‭Titus‬ ‭1:15-16‬

People can profess to know God, and yet by their very deeds they actually deny Him.

Same on Judgment day with Christ, people will say “Lord, Lord, haven’t we done many wonders and works in your name?” (Matthew 7:21-23) and yet, Jesus will reject them because He never knew them.

The only way to accurately assume the non-believers will get eternal life is if they are granted repentance and salvation by Jesus before the great day of judgment: (2 Timothy 2:25-26, Acts 13:48)

Those who are outside of Christ are “self seeking” so even if they visit people in prison or clothe people, if it’s not for Christ’s sake; it’s not the most pleasing to Him.

Matthew Henry’s commentary is helpful here:

The ground of this (Matt. 25:35, 36), For I was an hungered, and ye gave me meat. We cannot hence infer that any good words of ours merit the happiness of heaven, by any intrinsic worth or excellency in them: our goodness extends not unto God; but it is plain that Jesus Christ will judge the world by the same rule by which he governs it,

and therefore will reward those that have been obedient to that law; and mention will be made of their obedience, not as their title, but as their evidence of an interest in Christ, and his purchase.

This happiness will be adjudged to obedient believers, not upon a quantum meruit—an estimate of merit, which supposes a proportion between the work and the reward, but upon the promise of God purchased by Jesus Christ, and the benefit of it secured under certain provisos and limitations; and it is the purchase and promise that give the title, the obedience is only the qualification of the person designed.

An estate made by deed or will upon condition, when the condition is performed according to the true intent of the donor or testator, becomes absolute; and then, though the title be built purely upon the deed or will, yet the performing of the condition must be given in evidence: and so it comes in here; for Christ is the Author of eternal salvation to those only that obey him, and who patiently continue in well doing.

Now the good works here mentioned are such as we commonly call works of charity to the poor: not but that many will be found on the right hand who never were in a capacity to feed the hungry, or clothe the naked, but were themselves fed and clothed by the charity of others; but one instance of sincere obedience is put for all the rest, and it teaches us this in general, that faith working by love is all in all in Christianity; Show me thy faith by thy works; and nothing will abound to a good account hereafter, but the fruits of righteousness in a good conversation now. The good works here described imply three things, which must be found in all that are saved.

We also have to remember, for those outside of Christ, especially on judgment day, are spoken of as being in the flesh:

“Those who are in the flesh cannot please God.” ‭‭Romans‬ ‭8:8‬ ‭

Whenever an act is preformed by an unbeliever both now & up until judgment day; they don’t do it all to the glory of God, thus rendering their own works a displeasure to Christ.

“So whether you eat or drink, or whatever you do, do everything for the glory of God.” ‭‭1 Corinthians‬ ‭10:31‬ ‭

These same people on judgment day, knew of God’s reality and yet did not glorify Him as God. (Romans 1:21)

They didn’t do any deeds for the glory of God either.

It may also be noted that God is pleased with the charity and service that unbelievers do to a certain extent, but God looks at the motives of the heart. So when Christ comes back, He can justly say to the unrighteous:

Then he will answer them, ‘I tell you the truth, just as you did not do it for one of the least of these, you did not do it for me.’” ‭‭Matthew‬ ‭25:45‬

The quick answer is "No".

The Bible makes it quite clear that God does want everyone to be saved:

For this is good and acceptable in the sight of God our Savior, who desires all men to be saved and to come to the knowledge of the truth. For there is one God and one Mediator between God and men, the Man Christ Jesus, who gave Himself a ransom for all, to be testified in due time,
— 1 Timothy 2:3–6

But it also makes it just as clear that it is only through Jesus that anyone can be saved:

Jesus said to him, “I am the way, the truth, and the life. No one comes to the Father except through Me.
— John 14:6

“let it be known to you all, and to all the people of Israel, that by the name of Jesus Christ of Nazareth, whom you crucified, whom God raised from the dead, by Him this man stands here before you whole. … “Nor is there salvation in any other, for there is no other name under heaven given among men by which we must be saved.”
— Acts 4:10–12

This seems like a serious dilemma, and to many denominations, it is.

But if one studies the Bible and understands what God's plan is, it becomes obvious that this current age is not when God is trying to save mankind. (Even common sense tells us this: if this is the age of salvation, God has certainly not been doing a very good job of it.)

During these two thousand years, God has selected a very small flock to become his elect and receive salvation at Christ's return (at the first general resurrection). These elect will be transformed into immortal spirit beings and rule with Christ, here on Earth, for a thousand years (The Millennium).

At the end of the thousand years, there will be a second general resurrection. The vast majority of mankind will be resurrected into a physical life, be integrated into God's Kingdom, and that is when they will receive their first opportunity to be saved.

For more details, see my answers to:

• What is the Order of the Resurrection of the Dead?
• What is the Biblical basis for saying that we won't have a second chance to be saved after we die?

And without faith it is impossible to please God, because anyone who comes to him must believe that he exists and that he rewards those who earnestly seek him. (Hebrews 11:6)

I don't think you can find a verse that is plainer that that.