Religious institutions should choose who they want to marry.

Pedantic note. Religious instututions don't marry people, nor do state officials. People getting married marry each other, the role of the priest or minister or registrar or whatever at a marriage cermony is as a kind of official witness and master of ceremonies. Their involvement is in the last resort peripheral.

It does seem to me that getting all fussed up about the word "marriage" is a mistake. Rather as if it were suggested that banjos should be called guitars, perhaps to stop people making unkind wisecracks. Banjos and guitars have a lot in common, and some things not in common, and it's quite useful to have separate words to refer to them. Both can make great music together or apart.