Hi iBrian
iBrian asked: “…there's a lot of stock in Jesus being the Jewish Messiah - but, why does that matter if his teachings were for the gentiles, not the Jews, and therefore Jewish beliefs are not important? “
1)  I think It matters because the Judeo-Christian messiah and his teachings and covenants with him are for the world, not merely for a single group of people
THE PROMISE OF THE MESSIAH AND FORGIVENESS OF SIN ARE FOR THE WORLD AND NOT SIMPLY FOR ISRAEL
As the Midrash Sifri (Deut 343) points out, the Lord offered the Torah/Gospel to all nations. He is not merely the Messiah/Savior of the Jews, but rather he is the Savior of the entire world.
Thus the ancient Yahwist texts says, describing the plan of God as carried out through all generations and all peoples: “...and I saw Adam and his generation, their deeds and their thoughts; Noah and the generation of the flood, their deeds and their thoughts;....Abraham and his generation, their deeds and their thoughts......the teachers of the children in Israel and their generations, their deeds and their acts; the teachers of the children of the gentiles and their generations, their deeds and their acts;...all the prophets of Israel and their generations, their deeds and their acts; all the prophets of the gentiles and their generations, their deeds and their acts...And I saw: the Messiah the son of Joseph and his generation, and all that they will do to the gentiles..... 3rd Enoch 45:1-5;
This concept of world-wide salvation and not merely salvation of Israel is sprinkled throughout the Old testament era texts.
For example, Isaiah 42:6 says “ I will keep you and will make you to be a covenant for the people and a light for the Gentiles”. The word “Gentiles” in the masoretic here is “goyim”, (וְאֶצָּרְךָ, וְאֶתֶּנְךָ לִבְרִית עָם--לְאוֹר גּוֹיִם) referring to “non-jews. Similarly the LXX of Isa 42:6 says “και εδωκα σε εις διαθηκην γενους εις φως εθνων” (i.e. the “nations”, an idiom for non-israel).
Similarly, Psalms 2:8 uses goyim (i.e. the non-israel nations) as the Lords inheritance.  
“Ask me, and I will make the nations your inheritance, the ends of the earth your possession.”
The same is true of other pre-christian Yahwistic literature. The texts tells us the messiah is a staff “for the righteous ones” and not merely for israel. The texts reads:
“He will become a staff for the righteous ones in order that they may lean on him and not fall. He is the light of the gentiles and he will become the hope of those who are sick in their hearts. All those who dwell upon the earth shall fall and worship before him: they shall glorify, bless, and sing the name of the Lord of the Spirits. For this purpose he became the Chosen One; he was concealed in the presence of (the Lord of the Spirits) prior to the creation of the world, and for eternity. And he has revealed the wisdom of the Lord of the Spirits to the righteous and the holy ones, for he has preserved the portion of the righteous because they have hated and despised this world of oppression (together with) all its ways of life and its habits in the name of the Lord of the Spirits; and because they will be saved in his name and it is his good pleasure that they have life.” 1st Enoch 48:1-7;
Luke 24:47 also uses the Greek form for “the nations” (εθνη) “…and repentance for the forgiveness of sins will be preached in his name to all nations, beginning at Jerusalem…”
The ancient Judeo-Christian prayers reflect this doctrine that the Messiah is not merely the messiah of the Jews, but of non Israel as well. Hellenistic synagogal prayer #5 says of the Messiah: “For by him you brought the gentiles to yourself, for a treasured people, the true Israel, the friend of God, who sees God”.  Hellenistic Synagogal Prayers - #5
The apostolic Father Clement, who was a convert of the apostle Peter and co-worker with Paul also relates this same ancient principle, saying: “But of his Son the Master spoke thus: You are my Son; today I have begotten you. Ask of me, and I will give you the Gentiles for your inheritance and the ends of the earth for your possession.” 1st Clement 36:4
Thus New Testament Matthew reflects this same principle, saying : “And in his name shall the Gentiles trust.”Matthew 12:21
While most texts of Matt 1:21 say he will save his “people”, from their sins, the syrus Curetonianus variant of the New Testament says “for he will save the world (κοσμος) from their sins” (Matt 1:21)
iBrian asked: "…if Christians are not Jews, why would it matter whether Jesus fulfilled a Jewish prophecy or not?“
I am not sure that I understand this question.  Are you asking why the Christians care about Jewish prophecy of a Messiah?
IF this is your question, then I think it matters since the promise is that the Messiah would save all nations from sin and not just Israel and thus the promise of a redeemer applies to the Christians (and all others) as well.
Good luck iBrian in forming your own models regarding what these things mean and how to apply them.