Except it doesn;t. Right now, roughly 20% of all hospitals in the US are owned by a religion; most are Catholic, and about 1/4 of them are ‘some other religion’. That is up from 12% is 1995. What that means is that, in many cases–especially when it’s an emergency–you won’t have any choice at all except to accept religion-tainted healthcare.
I’ve lived in places where the only option covered by my insurance was religions.