Jesus of Nazareth (well, or of Bethlehem - seems pretty clear they swapped the birthplace around to make the 'facts' fit a prophesy) almost certainly wasn't born in December - but there were winter solstice celebrations in pagan (probably Druid with, depending on where, a mix of Roman Saturnalia) Europe.

Putting a competing holiday right on top was a way to move people away from paganism - otherwise they'd have simply added "the Christ's birthday" to their calendar - and continued to celebrate the solstice.

If it started out as non-Christian, then got taken over - why not take it over yourself?