Christmas is literally the celebration of "Christ's Mass." It is a eucharist in honor of His birth/incarnation. The eucharist is the table meal which connects us sacramentally (through faith and love) to the cross/sacrifice of Good Friday.
Christmas is, of course, the Christening of pagan holidays. Paganism predates Christianity and Judaism, no way around it. Trying to be free of anything pagan's did in the past is futile. Celebrating light and gift giving and love are pagan, but certainly not foreign to Christianity either. Sacred words and sacred meals are pagan, but the Bible and eucharist certainly have taken them to the fulness of their meaning. Truth be told, every Christian is a baptized pagan, so anything human is pagan, but potentially redeemable in Christ.
Keeping Christ in Christmas means that we remember that whatever its pagan roots, the season has been Christianized and its meaning changed for us who believe. Keeping mass in Christmas means making worship the center. Family matters. Gifts matter. Happiness and joy matter, but Christian and Christ centered worship matter more. It holds all the other things together.
Christianity is an "us/we" faith. We worship in community. That is why keeping mass in Christmas is so important.