The people are the government and the people should control marriage. Otherwise, anyone can claim to be a church and then marry gays or any other combination you can think of.
Someone could start a church called the Gay Church of Christ and marry all the gays they wanted and without a government regulation on marriage, there would be nothing we could do to control it.
Without government involvement in marriage, what would that matter? Anyone can create whatever kind of church they like, and call it whatever they want. This is established under the First Amendment. If someone wants to form a church that conforms to what they personally believe and conduct marriages, how does that affect anyone outside of whatever congregation they may have?
The people do control marriage. Obviously, one of the proper functions of government is to prevent people from being forced or tricked into marriages involuntarily.
If the government does not hand out goodies based on marital status who cares?
They could call it that; but it wouldn't be a church of Christ.