Evidently, Jesus links aspects of himself with aspects of God, at least in terms of status. I agree with Mac's Musings that one instance does not a godhead make, but Jesus often things like this. Hence, the disagreement between Trinitarians and branches that read Jesus as son or prophet, but not deity, lies in whether you take the sum total of all the times he does this as
- (a) a statement of identity with God
- (b) an ancient custom whereby someone's representative speaks as if they were that person
- (c) otherwise figurative language
In any case, the line of reasoning you're talking about is indeed attested. Similarly, some study Bibles and commentaries identify the various times one of Jesus's aphorisms or parables is actually one that was already told by rabbis, but he subtitutes himself for the God character. For example, there was a Jewish parable about building a house on the rock, but in the parable the rock was God's law instead of Jesus' words.
This is a very deep discussion, and it's not really possible to hash it all out here. In my opinion, it's also not possible to inarguably win either way, despite the confidence with which some people come down on one side or the other. It involves decisions about language and its literal and figurative uses. For example, C.S. Lewis cited Matthew 23:37 where Jesus uses the first person for a statement that only seems to make sense from God's point of view:
Jerusalem, Jerusalem, you who kill the prophets and stone those sent to you, how often I have longed to gather your children together, as a hen gathers her chicks under her wings, and you were not willing. (NIV)
Are we to take these words at face value and conclude that Jesus is speaking as himself here? Or is he paraphrasing the sort of thing God says through prophets in scripture? After all, if I say, "Et tu, Brute?" you understand that I'm alluding to Shakespeare and not that I'm Caesar or you're Brutus. Are we allowed to stray from the Bible's literal words and decide what is a rhetorical structure? Sometimes it seems reasonable, sometimes not. Too bad there were no quotation marks in Koine Greek writing. As it is, what we have are compelling arguments, not scientific facts.
This line of thinking is sometimes used to push claims of divinity earlier into the synoptic gospels, since John is generally seen as more explicitly supporting the identification of Jesus with God, but since it was written later some argue that it retroactively imposes theology on Jesus (another whole discussion). For example, in John 14:9, Jesus says "Anyone who has seen me has seen the Father" (NIV), which you can still argue is figurative but it's a harder case to win compared to the synoptic inferences.