The traditional chronology would have Zechariah's encounter with the angel happening at or just after the High Holy Days in the fall. John would then be conceived at the middle or end of October, so Elizabeth would be "in the sixth month" at the end of March (March 25, the Feast of the Annunciation) and Jesus would be born 9 months later on December 25th.
Priestly courses served two weeks per year, but not consecutively. They also served during major festival periods. Abijah's normal course of service falls roughly in the spring and the fall. According to this link, if you work back from the timetable Josephus gives us, you get Abijah's fall course ending on our date October 9, 6 BC.
Is that conclusive evidence that the traditional chronology is correct? Of course not. But it is evidence that it is not unreasonable.
The vast majority of scholars, good educated Biblical scholars put his birth in September to October.
You will be hard pressed to find any of the well known evangelical theologians who believe Jesus was born in December, let alone December 25th, which was adopted from a pagan holiday festival.
Simple response to your question is logic.
The reason Scripture does not mention it near a holiday is because the census forced them to be in Bethlehem. If it were for the reason of the holiday, Christ would have been born in Jerusalem, and not in Bethlehem as prophecy predicted it to be.