A large part of the problem here is that this contentious word "marriage" means two things that may or may not apply at the same time.
Firstly we have the spiritual concept of marriage as a covenant before God. Other faiths may have their own concepts of what "marriage" means before their gods.
Then we have the secular concept of marriage as defined by the secular government, in which two people take on secular responsibilities in exchange for secular benefits. Someone entering into such a contract need not have a faith in any gods, and given the secular nature of the benefits of this concept of marriage there is no obvious reason why any group of two or more people should not be allowed to enter into it. If, for example, my wife and I decided to enter into a secular contract with my best buddy and his wife, such that we could file one single tax return, we all had automatic inheritance rights and next-of-kin rights, we got automatic custody of their kids if anything happened to them etc, why shouldn't we be free to do so? As for who is having sex with who within the group, that's not really anybody else's business.
One fundamental problem over the years is that the church and the state seem to have gotten rather more intertwined than is ideal. If the church wants to offer a state-sanctioned service it can't really complain if the state then requires it to offer the service more widely than it might choose. It seems to me that perhaps the best solution is for churches to get out of the secular marriage business entirely and only offer a blessing service to couples once they are already legally married in a registry office. That way they can restrict the blessing service to members, and decide for themselves what membership criteria they want to put into place.
It's not a huge leap for such a thing to happen. In the UK there is at least one type of Islamic marriage service that is not recognised by the government, thereby requiring Muslim couples who want an Islamic ceremony to also have another secular ceremony where they take vows recognised by the state in order to be legally married.