Why couldn't deformed people be priests according to the Torah? (Leviticus 21:16-23).?

The Bible doesn't say, but IMHO it provides enough information to make a good educated guess.  Below is the key passage.

Ex 19:3 Then Moses went up to God, and the Lord called to him from the mountain and said, “This is what you are to say to the descendants of Jacob and what you are to tell the people of Israel: 4 ‘You yourselves have seen what I did to Egypt, and how I carried you on eagles’ wings and brought you to myself. 5 Now if you obey me fully and keep my covenant, then out of all nations you will be my treasured possession. Although the whole earth is mine, 6 you will be for me a kingdom of priests and a holy nation.’ These are the words you are to speak to the Israelites.”

It appears that one of the main purposes that God offered the covenant mentioned above to the ancient Israelites was to make them a "kingdom of priests".  A priest is a "go between" between God and man.  Thus it seems that it was God's intentions to use the Israelites as priests to bring the Gentiles to him.  Obviously Gentiles wouldn't be that interested in a relationship with God if his representatives were immoral OR (for some people) looked weird.

The issue was NOT with the deformed person, but rather how they were perceived by others in those days (and maybe even today depending upon the person).  For example, it was often thought that if someone had a deformity of some type, they must be on God's wrong side and that's why God gave them the deformity (ref Jn 9:2).  So to preclude that from being an issue, God made a rule against those people being his priests.

We see a similar rule when it comes to the qualifications of church overseers (bishops).

1Tim 3:2 Now the overseer is to be above reproach, ... 7 He must also have a good reputation with outsiders, ...

A Christian may be a very faithful good man, but might in his distant past have done something such that his reputation is tainted.  (That "something" may not even be a sin in God's eyes.)  Even if it was something sinful, God has forgiven and forgotten the sin.  But Christians are spiritually weak (Rom 14:1) and therefore struggle letting stuff like that go.  So that there are no issues like that to detract from the office of elders, God made a rule where such a person cannot serve as a bishop.

Even in the business world, business owners have to make decisions based on the "lowest common denominator" of customers.  Therefore some worker may be a very good worker, but has tats all over his face.  The owner may be hesitant about placing him in a position where he interacts a lot with customers because the tats would be a distraction for those customers who have an aversion to tats.

Back the Israelites; part of the deal God made with them was that they had to "fully obey" him.  Obviously if they did that, they would represent him well to pagans.  Unfortunately, within weeks they started disobeying him, to the point that about 900 years later, almost all of them had embraced paganism themselves, so much so that he called the whole deal off (Jer 31:31-32).  His plan for them as a "kingdom of priests" did not work out.

Because God required perfection for that one task

Harsh people, harsh law.  The people at the time were so hardened that they needed physical metaphors of spiritual unhealthiness.

God just liked the pretty boys.

Deformed people were considered imperfect, and imperfection was widely considered a sign that God deemed the person spiritually unfit is some way.

Was God unfairly discriminating against handicapped people when he said they were unqualified to offer sacrifices? Just as God demanded that no imperfect animals be used for sacrifice he required that no handicapped priests offer sacrifices.

The priests were a picture of Jesus Christ, who is perfect, so there was a picture type to be fulfilled.  Like also how it had to be a lamb without blemish, another picture of the perfection of Jesus Christ.