That's the exact answer. Churches can create new personal "covenants" not subject to government review if they wish, but they need to get out of the marriage business immediately.
Nobody cares if you do that, but people won’t do it, even George Washington was focused on making sure that his marriage was totally legal and not just a religious ceremony shared by him and Martha.
Not many people care to have a marriage that isn’t a legal marriage, for instance military people, what are they supposed to do?