The problem with that is what is stopping O from saying, “ok, you can have your marriage, but the state won’t recognize anyone you marry as married.”
If they are married in the Church - then they can’t get a civil marriage.
Granting your hypothetical of exclusion of church-only marriages, any marriage not conducted by an agency of the state still may be recognized through common law indicia of marriage. So while it would be tedious to not have a blanket presumption in place, specific cases already in court for other reasons (child custody, etc) would still be able to establish common law marriage.