I hear what you’re saying, and I definitely empathize with the view to just tell the government to go “stuff it”. But, stepping back and looking at marriage in the New Testament context - we must remember that the early Apostles were, outside of Israel, living in a pagan world. Pagan marriages were probably recognized, in general, even if they were not performed by Christian ministers or Jewish Rabbis.
It would have been very difficult for them to reach out to the pagan world while refusing to recognize ANY pagan marriages. The only stipulations that I see in the Bible where a marriage would NOT be recognized would be homosexuality or bestiality, etc. Paul says that Christians should not “be unequally yoked”, which by that he meant Christians should not marry unbelievers.
But, Paul does discuss the problem of husbands and wives who convert to Christianity, but their unbelieving husband or wife does not. He does not recommend that the Christian leave the unbelieving spouse - only that if the unbelieving spouse does not want to remain married to the Christian, then to let them go. This leads me to believe that the early Christians recognized pagan marriages as long as they weren’t violating a greater Biblical moral law (such as homosexuality, incest, etc.).
Therefore, I think that if Christian churches quit performing “legal” state recognized marriages, they should still recognize marriages performed outside of their respective congregations if that marriage does not violate the moral standards of the Scriptures.
The trouble is that this is a semi-legal problem, not just a religious one. US law is heavily weighted against “exclusivity”, for example, it is not even legal to sell or rent an apartment or home in an “exclusive” community, because this is seen as racist.
However, an exception to this weight is “membership”, which provides considerable protections in the law. But any organization that uses membership as a defense, must be very clear about who their members are, and what rules they must follow.
Churches, however, have allowed government to take over many of their formerly exclusive activities, like registering births and deaths. And, of course, marriage.
The trouble is that government continually tries to enlarge its power at the expense of religion. So they need to either take a stand, or forfeit their sacraments.