How do Trinitarians explain Jesus' response to the scribes and Pharisees at Luke 5:20-24?

The Trinitarian stance is that Jesus Christ was unarguably a man, “Son of Man” being his most-oft-used designation. It does not make sense for your question to add, “If the Pharisees were wrong in thinking Jesus was a man, why didn’t Jesus correct them, but instead reinforce that belief by calling himself ‘Son of Man’…”

No Trinitarian thinks the Pharisees were wrong to believe Jesus to be a man. Trinitarians also believe that Jesus was a man. He was fully man, but don’t be too hasty to add a full-stop there.

Second point about this question: There was nothing ‘simple’ about Jesus ‘just’ saying in public that he was God. For a start, that could be misleading (according to Trinitarian understanding of the complexity of the one Being of God). There is more to the one Being of God than the Word incarnate, made flesh, walking amongst other men who beheld his glory, that of the only-begotten of the Father (John 1:1-14). Jesus knew that if they didn’t understand that complex truth, they would instantly claim grounds for stoning him to death, but he had to complete his mission on earth. “My time has not yet come” he said several times during that mission. But when his time DID come, he went for it, telling those who hatefully asked him if he was “the Christ, the Son of the Blessed”, he said, “I am: and ye shall see the Son of man sitting on the right hand of power, and coming in the clouds of heaven.” (Mark 14:61-64)

This led to instant cries of “Blasphemy!” and they all condemned him as guilty of death.

That is why Jesus held back openly stating his deity until that moment, when his time had come. Stating that he was also the Son of God was too much for those who only thought him to be a man.

Third point about this question: Trinitarians understand why the Pharisees were offended at Jesus claiming (and showing) that he could forgive the sins of humans. The Pharisees only thought him to be a man. But had he only been a man, he could never have done the miracles he did, including raising the dead. So, to confound them, he gave that apparently indirect answer that stopped them in their tracks, forcing them to think about how he acted in conjunction with his claims, the former proving the validity of his claims to be able to forgive sins.

Jesus had to achieve the Father’s will before dying, and he knew that openly saying, “Oh, by the way, as well as being a man, I’m also God” would short-circuit the plan of redemption. Jesus had to arrive at the cross at just the right time, in just the right way. He did not enable the Pharisees to stone him to death for alleged blasphemy because he had to be hung on a tree to die, as accursed, the sinless sin-bearer.

It’s only when the Trinitarian explanation is applied that the full extent of the breath-taking enormity of just who Jesus is dawns, and then it becomes clear why it had to be God incarnate bearing our punishment for our sins that gets us on our knees in adoration. To say that it was only a man (even if a sinless man) dying for sinners makes no sense, for no man could ever achieve what Christ achieved – conquering death, and the one with the power of death, the devil – unless he were God incarnate. Christ “led captivity captive” in his train when he ascended “far above all heavens, that he might fill all things” (or, “fill the whole universe” NIV). Ephesians 4:7-10, which quotes Psalm 68:18, about God doing what Jesus did in the fulfilment, at his ascension. That is what it means to be Son of God. And that is why Jesus had to remain incognito, slipping in under the radar at his incarnation, and remaining discreet in what he disclosed to his mortal enemies, until his time had come – his God-appointed time to die, as a man, for sinners who would confess him as did doubting Thomas, “My Lord and my God!”

First, when Jesus uses the phrase "the Son of Man," He is using that phrase technically, as we use "the Messiah" technically (whereas it meant 'king' since it means 'the anointed one') — referring to the eschatological messianic figure of Daniel 7, who is worshipped by all nations forever as king, and which is pondered on in pre-Christian apocryphal Jewish texts under the same title ("The Son of Man"). This is why He uses the phrase in the third person, even though He clearly uses it of Himself, because He is using it as we use the term "the Messiah." 'Who do people say is the Messiah?' 'Messiah can forgive sins on earth.'

Notice that Jesus does not in any way engage with or challenge the assertion, "Only God can forgive sins," since this is self-evident. Note this first of all. Instead, He conflates the Son of Man with God (that is, makes the Son of God divine) when He says what paraphrastically equates to:

True, only God can forgive sins, but that it might be demonstrated that the Son of Man is God, He said before He healed the man, 'I forgive you your sins,' so that you would have no option in questioning whether God worked this miracle, since, indeed, who can forgive sins except God?

In saying, "authority on earth to forgive sins," He is saying that the Messiah was to show that He came to earth in His full authority as the incarnate Son of God, not as a mere messenger of the Son of God, but the Son, the Messenger or Angel of the Father; "Here, even on earth, I may forgive the sins of anyone I will."

Also, including the qualification "on earth" implies that the Son of Man was expected to have this power naturally, but that on earth might have been seen as unexpected (i.e. a divine figure that was supposed to leave His divine prerogatives in heaven, and be more 'human' on earth, which is here rejected).

His argument is that in performing the miracle of the gaining of the ability to walk of someone crippled and unable to walk, He proves whatever else He said, especially in the same instance, is true, namely, that He can forgive sins — since who grants the ability to walk to the lame, or forgives sins? Only God.

Saying that you have the authority to do something only God does is not proof you are not God, since God indeed has the authority to do everything God does...

In Trinitarian theology, Jesus is both God and Man, so the title "Son of Man" isn't incorrect in that regard.

Throughout Matthew, Mark and Luke, Jesus generally encourages the "Messianic secret" -- He doesn't directly state He is God. Instead He does or says things which lead sincere people to that conclusion.

This miracle relies on a somewhat subtle argument. The Pharisees accuse him of blasphemy for stating his sins were forgiven because only God can forgive sins. They aren't wrong on that point, but suppose Jesus were a fraud, He could say "Your sins are forgiven" which would, yes, be blasphemous as it would be claiming a power which belonged to God, but there would also be no way to prove or disprove that He wasn't God. Similarly if he stated, "but I am God," it couldn't be directly disproven.

Forgiveness of sin is not an observable act -- there is no physically discernible change that proves that Jesus did in fact accomplish what He said, so on the surface the Pharisees cannot distinguish the act of a blasphemer from God being present.

Hence, the miraculous healing. The argument thus becomes, given that Jesus can miraculous heal the paralyzed man, which only God can do, He must be God and thus can forgive sins.