As Apollo Creed said of Rocky: ‘whatever gets him in the ring.’
In this case, the objective is to get the Gaystapo OUT of the ring, by eliminating bureaucracy and paper shuffling as a proxy (in their minds) for wider social acceptance. They truly believe that a state-issued license (on which the word ‘love’ never appears) puts them on equal footing with real marriage.
The same principle applies to public school activities, dances, sports teams, etc. If it’s public, then leftists and militant gays will find a way to ruin it via legal attacks. The problem, of course, is that leftists are always attempting to make (or redefine) things as public even when they’re not - as the Indiana controversy demonstrates.
I believe gay marriage supporters opposed this because it leaves them without a fight.