Truly, every Christian has received at least one divine revelation: that Jesus Christ is Lord. That is not something man can arrive at by mental reasoning or by closing his eyes, covering his eyes, gritting his teeth and repeating over and over to himself "I believe."
And he said, Therefore said I unto you, that no man can come unto me, except it were given unto him of my Father. - John 6:65
Hereby know ye the Spirit of God: Every spirit that confesseth that Jesus Christ is come in the flesh is of God: - 1 John 4:2
But ye believe not, because ye are not of my sheep, as I said unto you. My sheep hear my voice, and I know them, and they follow me: - John 10:26-27
Amazing that some would think belief comes first when it is being born again that comes first.. the belief is ancillary.. and secondary.. Therefore some heretic's are/can be sheep but merely in error(possibly) about some things.. Belief can be altered or adjusted.. but there is no way to be UN-born again..
But, A-G, it is obvious that Peter did not really believe. If he did, he would not have denied Christ; he would not have started sinking on the lake, etc. He said he believed, but when challeneged, his faith failed.
If his faith was given, as you say, then his faith was also taken away. We have no control over it. If God gave Peter the faith to say what he is quoted as saying Mat 16, then God must have taken that faith away from him when he denied Jesus shortly thereafter.
But, the truth is, Peter did not say Jesus is God. We read into this for obvious reasons. In the context of the Jewish mindset of a simple fisherman, he said (stripped of all the hyped tag-names) "You are the anointed one, the favored one of the living God." He does not say that Jesus is God.
Those favored by the living God (i.e. son of God) was a title of angels (benei Elohim, sons of God, as in Job), powerful Jewish kings (of Davidian line), even powerful individuals, or simply those in "filial" relationship with God (adopted by Him):
Every Jewish king was believed anoinited by God. And Jewish messiah was supposed to be a Jewish king in the literal sense, and therefore anointed by God, not divine. Peter is simply stating that Jesus was that (future) king.
Many people think Jesus was condemned because He claimed to be the Son of God. That is patently false! He was condemned because He claimed that He had the power to forgive sins, which made Him equal to God, and that was blasphemy.
When you really think about it, the scenario is a mind-boggling: a poor carpenter from Nazareth convincing a dozen or so illiterate peasants and fisherman that He is really the savior king of Israel, and that they are His royal lieutenants.
It was probably believable until things started getting rough, then the whole thing fell apart. The real belief did not start until after His resurrection (a percieved miracle and not some divine "message" from God, downloaded into their hearts, as Paul claimed his).
But even then they doubted and wanted "proof." Thomas doubted aloud. But they all had their doubts. Not one of them is quoted as saying "Of course! I knew it" when the news reached that that Christ was no longer in His tomb.
What makes us Christians is an irrational belief (or hope) in bodily resurrection. But even Paul teaches to the contrary: "I tell you this, brethren: flesh and blood cannot inherit the kingdom of God, nor does the perishable inherit the imperishable." [I Corinthians 15:50]
This tells me that Paul believed (consistent with his Judaism) in spiritual resurrection, but not one of the body (this would also be very consistent with some other very Gnostic statements of Paul's earlier writings).
Excellent wisdom and Scriptures, of course.