Correct, he wasn’t born in December at all. For one the shepherds wouldn’t have been in the fields with their flocks in the cold.
But IIRC there’s something interesting that may have come out of use of the program Starry Night relative to a strong candidate for what was the Star of Bethlehem. As I recall from the presentation the date, by our current calendar, on which what might have been the star “stopped” over Bethlehem (Jupiter reaching the point at which it entered retrograde) as one left Jerusalem would have been on the day in question. Again, months after the sign first appeared and when the family was already living in a house.
Average December temperature in Israel is 60 degrees during the day. Remember, it's the Middle East.