All you say about Christmas is true. It is rooted in pagan tradition. But who cares about that? Christians have co-opted it for their own purposes and pagans can go on celebrating the solstice if they like.
The early Christians took a pagan holiday (the birthday of the Sun—December 25 was the time of the winter solstice at that time) and gave it a Christian meaning, just like they later converted some of the pagan temples into Christian churches (such as the Parthenon in Athens, the Pantheon in Rome, and the temple of Athena in Syracuse).