Popper knew this, by the way. So the inverse to the trap you point out here is the claims that "if not falsifiable, then unable to be true/useful", which is just as specious.
What systems for determining the truth exist that aren't built on top of some fundamental axioms?
my point is that "we define science as empirical falsificationism" is itself nonscientific by its own definition, and that that's OK, still capable of being true, and is a handy way to get stuff done.
but at this point i would say that "religion" is a very squirmy thing to define meaningfully without a lot of "well, but except when ..." clauses tacked onto the end.