Dec. 25 the date to celebrate Christmas?

I'd always assumed that there was no really good reason save political to have the nativity on Dec 25. Well it turns out that there may be more to it:
Chrysostom had a further argument that modern scholars ignore:
Luke 1 says Zechariah was performing priestly duty in the Temple when an angel told his wife, Elizabeth, she would bear John the Baptist.
During the sixth month of Elizabeth's pregnancy, Mary learned about her conception of Jesus and visited Elizabeth “with haste.”
The 24 classes of Jewish priests served one week in the Temple, and Zechariah was in the eighth class. Rabbinical tradition fixed the class on duty when the Temple was destroyed in A.D. 70 and, calculating backward from that, Zechariah's class would have been serving Oct. 2-9 in 5 B.C. So Mary's conception visit six months later might have occurred the following March and Jesus' birth nine months afterward.
I don't have the info to assess this, so I'm hoping that someone can give me a further pointer ...
Christianity :: Why is Dec. 25 the date to celebrate Christmas?