Well, what have we learned here then?
First, we know that festivals around the winter solstice are very ancient, much more ancient than Christianity. We know specifically for example that the Roman Dies Sol Invictus fell on 25 December as did the feast of Mithras, and we know that various characteristics of celebration, carousing and the like took place at these festivals.
We know too that several centuries after Christ the church, in the absence of any date of birth information from its holy texts, decided to pick the existing 25 December festival and to bolt its own celebrations on top of that, taking some rituals and symbols that were there already and adding some of its own. It also managed a pretty nifty re-branding job while it was at it.
We also know that many other characteristics were added much later on, especially by the Victorians.
The other thing we know is that for the most part people accept and enjoy this melange of influences and symbols as all part of the fun without ever needing to adopt the various religious beliefs that gave rise to them - paganism, christianity, Mithrasism etc.
So far, so jolly.
Then though we have the odd sight of BA doing his cuckoo in the nest act - not only does he deny the antecedence of Christmas from beliefs other than his own, he would also seek to re-invent the facts (about christmas trees for example) so as to co-opt them when they don't fit his thesis, and then to accuse others of hypocrisy for not embracing the superstitions he likes along with the cultural effect they had. In the absence of any argument to support him in this remarkably daft claim, he also quotes an atheist who happens to agree with him and in some as yet unexplained way just assumes that if other atheists agree with her about her atheism, they must also agree with her about anything else she may happen to say.
So here's the thing: the more pertinent audience for "why not leave 25 December alone?" isn't the atheists at all is seems to me, it's the BAs of this world. Of course, if ever he had the nous to recognise that accepting the addition of christian influences on the day is just good manners by the rest of us his whole problem would go away in any case, but I ain't holding my breath.