I think many of us Catholics had at the back of our heads growing up, unconsciously, at the very least that the idea of Christ's divinity, or His claim of divinity, was the hardest thing to prove. Some may even assume it was a fairly late addition to the story, like the half-hearted deifications of the Roman emperors. Much of this comes, not from history, but from historians, and rather bad popular historians at that. If one looks at the early Christological heresies, the one thing that it seemed nearly all Christians could agree on was He was emphatically not just a good man. Those outside the Church who denied His divinity, often thought Him considerably less than a good man, and there were many inside Christianity were not even sure He could be called a human being at all.
A lot of this comes from popularizers who find it convenient to throw out the text and tell us what really happened from their own speculations on human nature, or on their rather blandly respectful view of what they assume is Christian morality and their unfocused contempt for what they think is Christian dogma, as if the two could be separated. The Jefferson Bible, a bit of Enlightenment bowdlerization, is a prime example of this, in which Christ goes swanning around giving the impossible advice of the Gospels without the impossible miracles of the Gospel.
Anyone who tells you that they believe in the high moral system Christ developed, but not in His divinity, clearly has not read scripture. Christ was not a moral teacher in the manner of Confucius or Buddha, but came to fulfil the Law, doing so in startling and puzzling fits of drama and opaque parables that at times verge on performance art. Fig trees get cursed, Christ scribbles in the dirt, tells weird stories, asks people to eat him (which sent most of his followers running for the exits), uses scary phrases like people being "eunuchs for the kingdom of Heaven," and claims He knew Abraham some thousand-odd years earlier.
Chesterton called these the "riddles of the Gospel." Without the miracles and the resurrection, this is not so much a moral system as a cryptographic puzzle, and a rather disheartening one at that. Without grace (and the Eucharist), the Christian life is an unreachable ideal, and a rather Quixotic and bizarre one at that. It is not the self-evident Hallmark squishiness people assume it was--humility was never a virtue in the pagan world before Christ, for one thing. If you wish a high, human moral code, go and read Marcus Aurelius and contemplate your solitary stoic self-splendor, don't drag the God-Man into it. He is not telling you how to fix yourself, asking you to let Him do the heavy lifting.
The other thing is that it is clear to the early Christians who thrashed around with the tradition handed onto them in word and scripture thought this strange visitor was clearly more than a man. Christ's divinity was being praised in song as early as the letters of St. Paul (if Christ's own assertions to the effect in the Gospels are not enough), and it is clear from rabbinic commentaries of the period on Isaiah's prophesies that even before Christ came, it was thought the Messiah would have at the very least some special relationship with God, and at the most be quasi-divine Himself--He, the wonderful counselor, mighty God, the prince of Peace.
Indeed, the errors of early Christianity are quite illuminating in that regard. We have the Dan Brownian notion that the early heresiarchs were mild-mannered peacenik types who loved Christ the simple good man (and the goddess Mary Magdalene, don't ask me how that works), rooted in history and everyday life. With a few exceptions, by and large their Christ was not only scarcely human, He was scarcely historic. The Gnostic pseudo-scriptures show Christ as a weird ghostly being, not necessarily a god but a messenger from corporate headquarters sent to untangle the mess started by Jehovah, who in this view comes across as a sort of low-ranking Dwight Schrute weirdo in the greater scheme of things, with very little connection to Jesus the friendly aeon. Christ floats in, dispenses gnosis in a historic void lacking in the arguing Pharisees and Sadducees.
Why anyone would find this talkative, haughty spirit appealing is beyond me, but then I suspect a friendly ghost is considerably less demanding than a flesh-and-blood incarnate God. At the very least such discussions prove that the modern world's problem is not a lack of proof for historic high Christology, or its remoteness from modern man, but simply its inconvenience.