This discussion has aired some really interesting differences of opinion. As Larry has a delightfully direct way of expressing disagreements, most of my remarks will be directed to him.
I think we all agree that the archaeological evidence demonstrates that the Galilee was both Jewish and Hellenistic. Rabbinic notions of Judaism do not seem to have been especially prominent until the third century—with the possible exception of Judea.
I also agree that the modern scholarly search for criteria to judge historicity in the New Testament was at least implicitly anti-Jewish. Thank you for pointing this out. I think the field has self-corrected to a large degree. Still, the extent of Jesus' Jewish identity and background, especially in his sayings, has never been adequately assessed. It is an area in which scholars agree—yet ordinary Christians seem to have little awareness of Jesus as a Jew.
I absolutely agree that the baptism by John and the crucifixion are the two most important established facts about Jesus' life. A great many things can be logically derived from those two assured data. The baptism virtually assures Jesus' identity as a preacher of apocalypse and makes credible the idea that he could have been seen as a threat to the political establishment. Though Jesus' crucifixion is today seen as a badge of his martyrdom, it was a scandal in the first century and the church appears to have been sensitive to it in innumerable ways. This sensitivity is palpable in Paul. He reflects the concern that a crucified Messiah will be a stumbling block to Jews and folly to Gentiles. In today's Christianity, Paul's statement to this effect in Corinthians is almost incomprehensible. But to Paul, the problem was very real.
Paul, I think, continued to consider himself Jewish his whole life, despite the opposition he faced on that point from Jew and Christian alike. This brings me to Larry's insistence that I address the Jewish background of the infancy stories. Well, I agree to an extent that there is Jewish tradition in the birth narratives. Clearly, the Magnificat ode in the Gospels is based on Hannah's song in the Hebrew Bible. The early Christian or Christians who composed the ode knew Hannah's story. But that doesn't mean the Magnificat reflects a strongly attested tradition of singing songs of thanksgiving at the birth of a child in biblical or post-biblical Jewish tradition. There is a possible example from Noah's birth in the Dead Sea Scrolls, but the text is too fragmentary for sure conclusions. That's not to say I would mind being proved wrong here—after one accepts that the historicity is dubious, I don't see that much hangs on this either way.
The birth stories surely do express the specialness of Jesus' birth. But, on balance, they seem to me to use Gentile conventions to do so. The arrival of the three kings from the East is not specifically Jewish; rather, it demonstrates that purveyors of ancient wisdom acknowledge Jesus. The virgin birth is virtually unattested in Judaism. Divine fathers in births are totally unknown in Jewish tradition. What does occur in Jewish tradition is that God brings about the birth of special children by opening the womb of barren women. Isaac is the obvious example. But the barren woman motif is not prominent in the New Testament. In the Hebrew Bible, God also frequently puts his Holy Spirit on whomever he wants, often choosing neither the rich or the powerful—hence Jacob over Esau, David over Saul. It's true that there is an aspect of this tradition in Jesus' birth story. But it's hard to know about some of the other supposed connections. I find puzzling, for example, the stories' misunderstanding of the term "hosanna." I don't think that Jews would have used the term in place of "hallelujah."
Obviously, the messianic themes sounded in the birth stories are Jewish. But it is interesting to note how inchoate the messianic theme was in the Judaism of the time. There was hardly a fixed messianic job description (with well-attested birth stories) that Jesus fit. His messianic birth story is virtually the first one.
What explains the birth stories for me is the crucifixion. Only after the crucifixion is it clear to Jesus' disciples that he was a failed candidate redeemed by God. In this interpretation, all the Gospel prophecies of Jesus' redeeming death are vaticinia ex eventu—prophecies constructed after the fact. The messianic genealogy and birth are also better explained as a result of proclamation of Jesus' messianic status, following the crucifixion. So yes, the Holy Spirit is a Jewish tradition and the church is certainly expressing Jesus' life as a life governed by the Holy Spirit. But the means of expression impresses me as innovative as well as traditional.
Nor do I think that Jesus must have been born in Bethlehem. He could have been born there, but there is no strong scholarly reason to believe that he was. The omission of the birth narrative in Mark (and later John) explains a lot more than the birth stories do. In their effort to fill the gap, other early Christian writers used some Jewish birth conventions, but added startling details that are more Hellenist.
Ironically, perhaps, I think we agree on a lot more about Easter. If we assume the truth of the titulus, the charge on the cross, then Jesus was convicted of a political crime and executed as a political insurrectionary, whether or not this was an accurate view of his mission. Did Jesus think he was the Messiah? I do not find it necessary to posit that, though Jesus may have gradually come to the conclusion that he was expected.
What is clear is that the early church found Daniel 7 and Psalm 110, together with Psalm 8, to demonstrate the scriptural prophecy of Jesus' resurrection. With this came his messianic and divine status. The combination turned Jesus into Lord as well as messiah and allowed the church to identify Jesus with the divine name Lord in the Hebrew Bible. I think this was a Jewish process of hermeneutics—though there is no previous example of treating as prophecy this combination of verses and only limited evidence of humans achieving divinity in Judaism.
All of this only emphasizes the uniqueness of the events of Jesus' last days and the special regard in which his disciples held him. There are plenty of examples of Jewish expectation that the dead will rise on the last day, and that those who make others righteous will become angels or stars. What made Christianity special was its recognition that this process began with Jesus but had not been brought to fulfillment.
I might add that I find the celebration of Christmas to be particularly enjoyable—the music especially, but also the decorations and festivities. So I'm not being a grinch; I'm just reporting how I see the history. A Merry Christmas and a Chag Sameah to everyone. And a Happy New Year, too.