Yes, you are wrong. My wife and I married in a Church neither of us belonged to. We were interviewed by the pastor before getting permission to use.
Further, my old Church was regularly used for marriages by folks who had no affiliation to our Church or even our denomination.
The constant applied is the pastor of the church had to approve its use.
If we begin to have the government dictate the ceremonies which can be held at a Church, we may as well eliminate any notion that we have freedom of religion.
That probably means getting out of the "come get married here, we don't care about what faith you are" business.
The logical extension of this is to make Churches close their doors for any religious ceremonies unless they are members.
But if any organization of any type was actively advertising that it had a wedding chapel business, they are probably not going to be able to get away with changing policy. It would be like a restaurant fifty years ago where only white people would go by custom, suddenly hanging up a sign saying "no blacks allowed" after an antidiscrimination law had just been passed. It ain't gonna work. You can't force the restaurant to sell soul food, but if the customers want what's already been on the menu, you can't deny it to them. And saying, "Well, our hamburgers are not chitlins is not going to work in this situation, either.