I agree that there seems to be a sort of prejudice that only Christianity is a religion. Perhaps people understand on some deep level that Jesus and God are unique and really do belong in a different category from Buddha or a guru? Although it is the only holiday that mentions the name of the religious figure involved. I mean, there is no Moses Day, or perhaps we just aren't as offended if a holiday is named in a foreign language, probably because we don't understand it.
Even though there are more and more people practicing other beliefs, most Americans are still some form of Christian. We have a majority culture that is forced to deny itself in deference to assorted minorities. There is a lot of that in American politics these days. Now maybe there needed to be some changes to achieve parity, but we have just about managed a 180 on this. I wonder what happens when the majority decides they won't take it any more? Is it still a revolution when the people who supposedly have all the advantages are the ones who do it?
I will say though, I want my government to either acknowledge everyone's religions or nobody's. "Happy Holidays" for the month, or change the sign as required. Of course, Chanukah came at a slightly different time this year. Solstice and I think Yule were the 21, Christmas is the 25, Kwanzaa starts the 26, New Year's on the 1st. That's a lot to put on one sign for school break. Probably someone got lazy and went for the sort message.
I understand the best guess is that Jesus was born some time in the spring. Apparently the shepherds were most likely to be hanging out on the slopes together during lambing season. Otherwise, there was probably only a youngish child out there in case of disaster. The symbolism of Jesus being born at this time is a little more complex.
Most pagan religions celebrate the Solstice, which is the day of the year with the longest night and shortest day. They usually have both a god and a goddess. The goddess tends to symbolize something eternal that gives life (birth). The god is different between traditions, but in cultures where the god was associated with the sky or the sun, this is the time of year when he is born. That is, this is when light returns to the world. So there were already lots of people celebrating pagan holidays at this time, and the church needed to compete with that. Of course in it's earliest form Christmas really was a Christ Mass, but it reminded Christians who they were and gave recent converts something to do on a day they were used to celebrating already. Not to mention that in an agrarian culture, there isn't much to do in the winter, compared to spring planting or fall harvest, so there is time and opportunity for observing holidays.
I also understand that the Romans had 2 festivals around the new year that relate to Christmas. One was Juvenalia or the feast of youth. People gave presents to the children or youngest people in the family. I forget the name of the other, but it was essentially the same thing for elders. They were paired to represent the old year and the new year. So when we talk about secular Christmas and Santa Claus being for the children, we're not far off from some of the roots.
So the early church adapted local customs to its own purpose and added a Christian gloss to many holidays people celebrated anyway. These days we have grown up with the theological reasoning behind these observances and our culture no longer keeps the old holidays, so we don't even think of it.