Zeuhl, it may be more complicated than that. From Wiki:
The date of Christmas may have initially been chosen to correspond with the day exactly nine months after early Christians believed Jesus to have been conceived, as well as the date of the southern solstice (i.e., the Roman winter solstice), with a sun connection being possible because Christians consider Jesus to be the "Sun of righteousness" prophesied in Malachi.
In other words, the Annunciation - tied to the spring solstice - came first and Christmas was calculated on from that.
I've never been sure why a Rome-based church should celebrate with yule logs, hardly a feature of Italian life, and I wonder if north European pagans may have imported their old symbols into the Christian festival rather than the other way round.