Christmas, in and of itself, NEVER started out Pagan.
That time of year was traditionally reserved for Pagan celebrations (Winter Solstice, etc.) but the Christmas celebration itself, was not Pagan. Ever.
The celebration of the birth of Christ was moved to the end of December to rival the Pagan celebrations going on at the same time, so that the Christian Church would have something to celebrate rather than be tempted to join in the Pagan celebrations. The Church decided that Christians, rather than joining in Pagan celebrations (or just sitting at home), should set themselves apart from The World, and celebrate the birth of Christ Jesus. And amen to that!
No one ever believed Christ's virgin birth occurred at the end of December. That's just derogatory anti-Christian rhetoric aimed to make Christians seem ignorant.
The whole point of the Christmas celebration is simply to observe and be grateful to God for His taking on human form to offer us the gift of Grace and redemption. The time of year it is celebrated is completely immaterial, and therefore it was strategically marked on the calendar during a yearly Pagan Solstice celebration. A win-win, you might say.
"But let us not come up with any patronizing nonsense about Jesus being a great teacher. He has not left that open to us. He did not intend to." C.S. Lewis