The church analogy has nothing to do with this. There isn’t a consitutional provision requiring that someone be born a certain religion. I will say however that Chritianity is somewhat unique in that lineage doesn’t determine religious identity. For Jews and Muslims, if you are born a Muslim, you will die a Muslim, it doesn’t matter what you do in the interim, and any children born to either the father (Islam) or the mother (Judaism) is Muslim or Jewish respectively, again, regardless of what the parent may or may not have done before or after the child was born.
But back to the point, according to your theory it does matter how Canada treats my wife’s U.S. citizenship, because if Canada continues to recognize her as a citizen, and thus my son also as a citizen, then he would not be eligible, but if they don’t recognize her as a citizen, and thus not my son either, then he would have been born owing no foreign allegiance, and would be eligible. This is why that can’t be the rule, otherwise Canada gets to determine who’s eligible to be President of the U.S.