Yep the birth of Christ just coincidentally coincides with the end of Brumalia, which of course noone noticed when the emperor suddenly insisted everyone become Christian and had the bible written by committee. And it's of course a coincidence that that was (back in the day) exactly the winter solstice. And it's also just a coincidence that Jesus' life story has quite some parallels to that of earlier sun gods from the general area.

Most current Christmas traditions are more Germanic in nature, though, e.g. the Christmas tree. While in the current form a quite recent invention, decorating the house with evergreen stuff was common through the ages -- branches, wreaths, not whole-ass trees. The needles btw are fine smudging material don't just sweep them away.