You are close. And I don't think (most of the) people here dispute there is the religion part here and the... Uh... *Other* religion part here. Only the the mouth-breathers don't get that there are two different things going on here. But the fact is, the "birth" was moved. And it was moved to co-opt a previous celebration by a different religion. It's not "to make pagan traditions less secular and more palatable" (whatever that last part means). True, it does diffuse some of the pagan ritual, but it also adds credence to the religion co-opting it. "That thing all of you are celebrating? Surprise! You are actually celebrating our belief system! See, we even changed the name to prove it!"