You're certainly correct, Alan, in pointing to the particular matters that pose difficulties in treating the Gospel stories of Jesus' conception and birth as simple "historical" accounts. I also agree, of course, that there are difficulties with treating as simple history a good many other stories in the Bible and in the dearly held traditions of Christianity and Judaism (such as Hanukkah). But, as it's nearly Christmas Day, let's focus a bit further on some other issues to do with Jesus' conception and birth.
The first thing I want to underscore is that there are really at least two major questions to bear in mind. We could ask how to approach the tradition of Jesus' miraculous (virginal) conception in terms of historical inquiry: How might historians engage the matter? Or, we could ask whether and how the claim of Jesus' virginal conception is meaningful theologically: What is its relevance or meaning within and for Christian faith (for believers), or for understanding Christian faith (by observers of Christianity)? Let's start with the first question.
The earliest clear assertions of the idea of Jesus' miraculous conception are obviously the two accounts in the Gospels of Matthew and Luke. Now, these two books are widely thought to have been written sometime between A.D. 75 and 90. The very differences between the two birth narratives that you mention, Alan, make it difficult to derive either from the other one. Yet, along with the differences (shepherds, angel choir, census, murder of the Bethlehem infants solely in Luke; and Magi, star, Joseph's dreams solely in Matthew *), both accounts present Jesus' conception as miraculous. This suggests, perhaps, such a tradition dating back earlier than either of the Matthew and Luke narratives—perhaps years earlier, perhaps more.
Also, I find it interesting that both narratives are thoroughly told with reference to biblical and Jewish traditions, themes, ideas, aspirations, and hopes. There may well be very subtle allusions that one can infer to non-Jewish ideas, but the overt references all are to biblical stories and cognate traditions. For instance, the famous Magnificat, the ode in Luke 1, is quite obviously based on the song of Hannah (1 Samuel 2). Does this suggest that the claim about Jesus' miraculous conception—and perhaps even the gospel narratives as well—emerged first in early circles of followers of Jesus who were either Jewish ("Jewish Christians") or at least were appreciatively familiar with such circles? If so, once again, the birth story would have an early provenance, and one for which "pagan" traditions about women impregnated by deities wouldn't count for much.
Of course, early doesn't equal factual. All I'm saying is that the history of the conception story may tell us a lot about where that tradition comes from and why it emerged. For something to be "historical" in the usual sense of the word, it must be explainable on the basis of what we know and can show as operative in the world in some observable way. And the key "problem" with the claim that Jesus was miraculously conceived is that it emphatically asserts a unique event that is not to be understood as the result of some ordinary or observable force operative in similar events. So, it seems to me that all that we can do as "historians" is to study the history of the claim, the traditions, and the sources in which they are preserved. All that is "historical" in the sense that I have used the term here, definitely. But was Jesus' conception "historical" in that sense? No. That doesn't mean that it can't have happened or didn't happen. I'm not saying that. I'm only saying that Jesus' miraculous conception can't be presented as something to which consent can be compelled by force of historical argument, in the way that we can defend a claim about more mundane events.
Although moderns are terribly exercised by the question I've been discussing (all too briefly), I think the more promising question is what the stories about Jesus' conception and birth meant for those who transmitted them, and what the idea might mean for other Christians (and those interested in what Christians believe). At the very least, we have an emphatic claim that Jesus—and all that he means for Christian faith—is to be understood most clearly as deriving from God's initiative. That is, the Jesus of Christian faith is not simply the predictable product of a given set of social and historical circumstances. He is at the most important level a divine gift and is to be understood in terms of the character and purposes of God.
I'll conclude (for now) by agreeing with Alan on one more thing. Whatever we are supposed to make of the stories of Jesus' conception and birth, arrogance and religious self-congratulation have no basis in these familiar narratives. I hope that's not banal. Heaven knows it's a point that bears repeating. But I've found that the most devout, and those who have drunk most deeply of their religion, rarely need reprimanding about arrogance. Instead, it's most often those with just enough religion to make them dangerous! So, whether as a celebrant or an observer of Christmas, the stories point us to something worth celebrating, but in humble celebration.