@Joseph
Not everything in law is reduced to what is in the statute books. Sometimes it is just by precedent. Most states recognize marriages performed by pulpit Rabbis, Ministers, Priests and Mullahs. They are permitted to sign the civil marriage licenses as the officiant. They do not recognize most of these people if they do not hold a pulpit or the pulpit is out of state.
I’m a Justice of the Peace. When last Ms. CTL was married here at our home this August, the Mesader Kiddushin was from South Africa. He could not sign the CT marriage License, so I signed it.
See my post about my parents who married in Manhattan with only a Rav as Mesader Kiddushin, not a marriage license. Their marriage was recognized as legal