If you want to have your own religious (or non-religious) marriage do it, you always could, complying with law, or the “state” is voluntary.
You don’t think that Mormons, and Muslims, and gays, and atheists, and anyone who wants to, have been conducting private marriage ceremonies all along, meaningful to themselves, without caring that the state doesn’t recognize the marriage as legal?
Do whatever you want, if you don’t want to go get a license, or seek legal status, then don’t.
Sure, couples could have their own ceremonies and commit to each other.
But, according to what I’ve read, if two people want to be married in a church, the person officiating the ceremony is required to also report the sacramental marriage through the state as a “civil marriage.”
I did read that, when couples are in the country illegally, sometimes a Catholic church will marry them secretly, but they’re not supposed to.