Getting married at a church does not get you a marriage license.
The "legally binding" part comes when the officiant has the legal authority to conduct a marriage ceremony. Most pastors are licensed to conduct marriages. But if a bride and groom (or two brides or two grooms, whatever) get married by a chimpanzee at the biggest church in Houston, they are still not legally married because the chimpanzee (probably) doesn't have the legal authority to marry them. It's not location location location.
The most compelling argument I have seen lately for calling it marriage versus civil union is for purposes of adopting children. Only a legally married couple can adopt in some states. So if two lesbos have a turkey baster baby, and raise it together, only the biological mom has parental rights over that child. The other "mom" can't make medical or financial decisions related to the child, and can't have visitation rights if they split up. Married people and biological parents have those protections under the law. Their argument is that the non-biological parent is just as much a parent as the biological one, and should have the ability to legally adopt the child and become an official parent in the eyes of the law.
I don't think you are going to have as much trouble with churches as you think. Many churches have rules about who is allowed to marry in their church. Mormons won't even let non-Mormons *in* their church. I couldn't get married in a Catholic church by a priest because I am not Catholic. So it seems to me that people can complain, but oh well. Churches get exemptions from laws that private folks don't:
Gonzales v. O Centro (2006): Unanimous SCOTUS ruling in favor of the União do Vegetal (literally, “Union of the Plantsâ€), who sought an exemption from federal drug laws so they could perform rituals involving the hallucinogen hoasca.
Native Americans can own and gather bald eagle feathers for use in religious ceremonies, possession of which for other people is a FELONY. You and I can't even pick up a dead bald eagle off the side of the road.
SCOTUS has also unanimously ruled that churches can hire and fire clergy and other staff members without regard to anti-discrimination laws. Old geezer preacher? Fired. Baptist guy who decides he wants to be a woman? Done.
SO if a Baptist church doesn't allow a gay couple to get married there, oh well.