No, they don't. That's my point—all the bits that make up secular Christmas have Christian origins and have become secularized over time. For every tradition that is commonly cited as having pagan origins, we can trace it back until it becomes entirely unrecognizable and it's still all Christians all the way down.

As near as I can tell the myth of the pagan origins of Christmas has its roots in fundamentalist Christians who wanted to abolish things that aren't contained in the Bible. "Pagan" made a good rhetorical whip at the time, but it's since been taken as a serious approach to history by popular culture.