remove the government's influence from the social/spiritual/religious institution of marriage.

The government should only hand out civil unions rather than marriages, as it can and should regulate the legal, but has no place in the spiritual.
What any church or individual believes is not the government's business, according to separation of church and state. Moral judgements lie purely in the realm of the spiritual, and we gather in communities according to how closely that organization's beliefs match our own. If they suddenly stood for something else, we would choose a new community.
A couple could get a civil union, for all legal needs. A couple could get married, standing before their loved ones. Or they could get both. The government would have no say in who gets married, by whom, or how.