Doesn't that mean that the church won't marry you if the government hasn't made it official?

That just follows the argument that the religious aspect is simply a voluntary add-on to the actual institution of marriage (a government-approved contract between individuals), which is the opposite of what the original commenter was arguing for.

Matrimony and marriage are separate already. If you want to get church-married, go for it. If you don't, don't. Doesn't sound hard or like it needs to be changed.