I don't see the dialogue about the Son of God at the end of "Bread and Circuses" as an endorsement of Christian belief. If anything just the opposite. This planet, with a parallel evolution to Earth's, had a Caesar and a Christ, which suggests that Christ is a historically determined role played by multiple people on multiple worlds, rather than the one and only Son of God. Granted, the "Son" worship of the Roman planet is portrayed as socially beneficial, but not necessarily theologically genuine.
And Christmas, like many other holidays with religious origins, has already become as much a cultural celebration as a religious one and might be further secularized in the future (as other holidays like Halloween already have been), so I don't see a Christmas celebration as being very telling about what the crew believes.
Maybe the crew of the Enterprise aren't a group of doctrinaire atheists, each and everyone, but I do think a strong strain of atheism and secularism runs through their thinking. There's certainly no evidence that they're a particularly religious bunch.