Not to sound contrarian, but honestly, I think it would probably be better for our interpretation of Matthew to think of Jesus as a rabbi/ catechist. Reading Matthew to try to reconstruct its author is a bit like reading 'The Scarlet Letter' to learn about Nathaniel Hawthorne, without knowing who that is, and then going back and reading his book again to see how your hypothetical person would have written the thing – I mean, yeah, you can do it, but it’s a bit off.
But, yes, Matthew’s author certainly seems to have an interest in rabbinic teachings – both to borrow their pedagogical style and to depart from their tradition. An ordinary reader, especially if reading one of those antiquarian versions of the Bible with Jesus’s words in red, would soon notice that Christ’s teaching are lumped together here in a way much less predominant in Mark and Luke. It starts to look like five discourses with travels and miracles in between, with the denunciation of Jerusalem being the beginning of a Matthean climax.
They’re the flavor of the gospel, if not precisely its purpose. Jesus uses rabbinic language: “but I say to you that everyone who is angry with his brother shall be liable to judgment. Jesus repeatedly talks about revealing things, Jesus begins preparing his disciples for Jerusalem after the Pharisees come seeking signs and even his disciples do not understand him, and Jesus teaches his disciples after his resurrection once again to that they will comprehend the scriptures. He even gives to Peter the power of binding and loosing, the pseudo-rabbinic practice of deciding which laws are optional to follow and which are not.
But to say that Jesus and/or Matthew’s author is a rabbi is not enough; there was obviously more than one kind of rabbi in first-century Palestine. He’s not like the Baptist, who “proclaims threats of damnation, repentance, flight from the world.” He’s obviously not like the Pharisees, who he condemns in Jerusalem. He’s obviously not the Sadducees, who don’t believe in an afterlife and only hold Torah as scripture. Indeed, his path swerves abruptly from theirs, as he is “deprived of his mission’s success amongst his people by the Jewish peoples’ and their leaders’ own guilt” (Dobschutz 34).
What kind of rabbi is Matthew’s author, then? “The gospel...is best understood as a mirror of the competition and conflict between the Christian Jews for whom Matthew writes and formative Judaism, the movement that eventually evolved into Rabbinic Judaism” (Hare 408). This fits the Judaism we know from Matthew’s rough dating after 70 CE, the trend of Judaism and the Jesus movement in the Synoptic Gospels themselves, and a certain psychological sense: change comes from within.
The harshest critics of a movement often come from disillusioned proponents of that movement. This framework might even make more sense of the somewhat muddled Christology: the Matthean tradition might have been genuinely breaking new messianic ground. Novelty is often composite, so it might be reasonable, if not exactly logical, that Jesus would be “a human, divinely empowered Messiah, whom God will one day install as king and judge” – and one distinctly flavored by a wisdom and rabbinic tradition, and certainly worthy of worship (Hare, 410).