Proof is also found in the claim of the early church fathers that Matthew's Gospel was originally written in Hebrew [Hebraisti], then translated into Greek and then into Aramaic [Syriac, specifically Peshitta Syriac].
If no Jew spoke Hebrew or understood it, then how was he able to accomplish the task or why would he have done so in Hebrew if no one was going to be able to read it.
To distinguish it from the Chaldean Aramaic. The appearance of Aaramaic goes back as far as 8th century BC. During the Babylonian domination, the Jews began to use Aramaic in official communications and, little by little, it became the dialect of the Jews in Israel, despite considerable resistance of the religiou leaders.
The particular Aramaic dialect spoken by the Jews was the Chaldee (or Chaldean) Aramaic (which the Greeks called hebraïstí), not Syriac (which the Greeks called syrïstí). In the OD, the Greek LXX (Septuagint) refers to (biblical) Hebrew ioudaïstí.
The Jews in the first century AD did not speak Syrian Aramaic but Chaldean Aramaicnot (biblical) Hebrew. Although their differences may have been greater than those in the modern English versions, it is the same as saying that Americans don't speak British English, but American English, but it is English nonetheless.