Isn’t the reason that the church makes such a big deal of repeating this whole “Christmas day is when Jesus was born” and “Jesus was born on Christmas Day” due to exactly the point that it wasn’t the day he was born.
The church moved the celebration to better match the pagan celebrations, so they needed to push this lie a lot to get people on board with the fact that they moved the birthday to make the sales pitch for Christianity better match the pagan traditions.
You would have though that the god of all existence could have timed the birth of his divine child a bit better, but I guess we all know that it’s easy to sometimes miss deadlines and be off by a month or so. Also the biology of virgin pregnancies, though unexplored scientifically likely followed the biology of regular pregnancies, so there might be a bit of uncertainty on the exact time of arrival even if god did impregnate Mary with the intention that she’d give birth around what is today Christmas Day.