So you want to do away with Marriage Licenses and leave it to religious institutions to decide who should be married and who shouldn't? Fuck them, they should have no say in it what so ever beside the ceremonial events they perform.
Honestly there's nothing wrong with that. Since, if we do away with it, there's absolutely NO difference between a married person and an unmarried one. Hell, people could marry 20 people, their pet goat, and the corner mailbox. It wouldn't have ANY meaning so there'd be no reason to care. In fact, we'd save trillions on divorce proceedings.