If Christianity has always been a cafeteria religion, then people who want to get as close as possible to the original Christianity (as evangelicals usually do) won't be worried by accusations that the original Christians picked the bits of Judaism they liked, because as far as evangelicals are concerned, the original Christians had the authority to do so. So the Bartlet gambit is no help when arguing with them.

I don't think calling Christianity a cafeteria religion is that useful, because in the sense you're using it, all religions which started by modifying a previous religion are "cafeteria religions". The "cafeteria" term usually refers to people who say they're part of a well-established religion but don't follow some of the major doctrines, not to the establishment of new religions.