You have a different take on history than I do. In my view, rabbinical Judaism was established after the destruction of the temple, a development of certain strands of the Pharisees.
A woman of unknown ancestery? Galilee was a frontier region for the Hasmonean government. It may be speculated that Joseph was a native of of means who had relatives in Nazareth, of whom Mary may have been one. In any case, Mary was a woman of Israel, otherwise Jesus was not a Jew, which is, of course what you choose to believe. It is strange therefore that he limited his ministry to the Jews, with the exception of his mission into Samaria. He did say that salvation is from the Jews, unless you are trying to spin that as an ironical statement.
If Rabbinic Judaism only came after the destruction of the Temple, how come then people called Jesus "Rabbi" (cf. St John 20.16, etc.), and how did he come to preach in Synagouges (St. Luke 4.15-21) that held services which are described in a form an Orthodox Jew stepping out of modern New York or Tel-Aviv would recognize and could probably immediately join? Modern Rabbinic Judaism was already then extent.
Galilee was a frontier region for the Hasmonean government. It may be speculated that Joseph was a native of of means
If St. Joseph was from Bethlehem (St. Luke 2.3), which he obviously was if we believe the Gospels and Prophets, he could hardly be from Galilee. In fact, it took a dream to convince St. Joseph to give up settling with his little family in his native Judea and move to Galilee (St. Matthew 2.22)
In any case, Mary was a woman of Israel, otherwise Jesus was not a Jew, which is, of course what you choose to believe.
Jesus was not a Jew, since he was not from Judea. Rather, all through the Gospels we read about how the Jews were trying to kill Him. Mary was a woman of Israel and of probable mixed ancestory since she was from Galilee of the Gentiles, where the Israelites had been promiscuously mixed with the Gentiles (Hosea 8.8) to the point where the Talmud rejects them as Jews.
It is strange therefore that he limited his ministry to the Jews, with the exception of his mission into Samaria.
No, Jesus preached all over the region, also going to the area of Tyre and Sidon, spending much time in Galilee itself, and going across the Jordan.
His ministry was not to the Jews, but to the "lost sheep of the House of Israel". The "House of Israel" of course being the so-called Ten Lost Tribes, since the Jews were the "House of Judea" and were not lost at all. Of course, this was not a measure of exclusivity at all. Among his Apostles was Simon the Cananean, and he certainly ministered to many Gentiles once they demonstrated their faith to Him. How could He not, since the Prophecies He had inspired in Isaiah foretold exactly that?
He did say that salvation is from the Jews, unless you are trying to spin that as an ironical statement.
Meaning salvation is from the faithful of Israel, who were given the divine message of God's revelation and the blessings and entrusted to pass it down for all men. But temember "For he is not a Jew who is so outwardly in the flesh; but he is a Jew who is so inwardly, and circumcision is a matter of the heart in the spirit, not in the letter." (Romans 2.28-29). And also be not taken in "by those who say they are Jews and are not, but are a synagogue of Satan" (Revelation 2.9) seeing as they have not Abraham, but the Devil for their Father (St. John 8.42-44), even though Jesus acknowledges are genetic children of Abraham (St. John 8.37).
Also remember, as St. John Chrysostom notes in his commentary here, he is not speaking of who He is, but of who the woman thinks He is.