You've touched on two topics, that I wanted to comment on.
Aramaic in the NT
This is an interesting area that I don't know much about. I hope there will be further discussion on this topic.
There are a couple of instances where it makes sense that it comes from Aramaic or Hebrew. Bearing in mind that the two languages were so similar that most of the time if someone tries to nail it as exactly Aramaic or exactly Hebrew, they're overstating it.
Edit: in modern terms, imagine I had a halluci... err... vision, and wrote about an Italian messiah, based on some phrases out of context... err... "prophecies" in the Vulgate. For some things you might be able to tell whether my English phrase can't come from modern Italian, or can't come from classical Latin, but for a whole lot of stuff you wouldn't be able to really tell which. That's pretty much the problem with Hebrew vs Aramaic.
But it's kinda important to remember that, because a lot of claims boil down to, "it comes from Aramaic, therefore Jesus said it", which, in a addition to the obvious problem, is begging the question in identifying the language as Aramaic. The reasoning assumes that it was said in Palestine, where Aramaic was the language and Hebrew was just a dead liturgical language, as opposed to coming from some non-canon "prophecy" in Hebrew that can't originate with Jesus.
Anyway, probably the most obvious is the "son of man" phrase which later the church took as a title for the Messiah. (And we even saw someone on this forum claiming that Jesus was claiming a greater title with that than even son of god

) But actually in BOTH Aramaic and Hebrew, "son of man" was just a sort of kenning for "man". They used it for more than "man" though. E.g., "the son of a donkey" construct worked the same.
So for example when Mark makes Jesus say, in Mark 2:27-28:
27. Then he said to them, "The Sabbath was made for man, not man for the Sabbath.
28. So the Son of Man is Lord even of the Sabbath."
It really makes no logical sense. But in Aramaic or Hebrew it would just mean "The Sabbath was made for man, not man for the Sabbath. So man is Lord even of the Sabbath." The "son of man" was just a literary device to avoid repeating "man" ad nauseam. Ah-ha. Now it makes sense. But it suddenly stopped being some claim about Jesus's authority over Sabbath, and it's just a generic way of saying, "screw this, it's ok to break the Sabbath."
That said, the criterion is often abused for Jesus alone to mean something it really doesn't: support a character as true.
Actually the criterion is a solid and rational one to EXCLUDE stuff that couldn't have been said in a particular language. E.g., John's "born again" conversation is based on a pun that only worked that way in Greek, and would definitely not worked that way in Aramaic. So we can be pretty sure that it wouldn't have been said in a conversation in Aramaic.
The criterion is used to flag something as probably FALSE, not to support it as probably TRUE.
The reason is pretty obvious. Using an expression that came from another language, doesn't actually mean that one can narrow who said it, or even that it was actually said by anyone else than the author writing that stuff. E.g., just because Pitr from the User Friendly web-comic uses the occasional Russian words or stereotypical Russian constructs, doesn't mean that there was a historical Pitr actually saying that.