Prior to the exile we don't see the term Jew.
So while your comments make sense, they do not account for the absence of the use of the term. In fact, outside of Nehemiah and Esther we barely see it at all. We never see it in Daniel.
There is the famous line in 2 Kings and Isaiah
Speak in Judean, which I guess people figure is Hebrew, apparently no one having the slightest idea what else that could mean.Eliakim, Shebna, and Joah replied to the Rabshakeh, "Please, speak to your servants in Aramaic, since we understand it; do not speak to us in Judean in the hearing of the people on the wall." (Isa 36:11 TNK)
I'm not sure what this does about the many interesting points you raise, but I'm not sure we can speak of the term Jew in the same way as Israelite and Hebrew.
PS. An odd example is that the word appears once in Zecharia
That's an interesting place because it comes in just before Zecharia 9.Thus said the LORD of Hosts: In those days, ten men from nations of every tongue will take hold -- they will take hold of every Jew by a corner of his cloak and say, "Let us go with you, for we have heard that God is with you."
(Zec 8:23 TNK)
Zecharia
My hero Konrad Schmid might date this to the Ptolemaic period - at least Zechariah 9-14.Zechariah 1–8, sometimes referred to as First Zechariah, was written in the 6th century BC.[6] Zechariah 9–14, often called Second Zechariah, contains within the text no datable references to specific events or individuals but most scholars give the text a date in the fifth century BCE.[7]