Something like this was proposed 50-some years ago by C. S. Lewis who was commenting on the deteriorating state of marriage in England in the mid 20th century. There would be church marriages and government “marriages.”
I’m looking at how well the government has lately been serving as a steward of the institution of marriage, even if all the “gay marriage” stuff could be subtracted out of the picture. And the record is atrocious. While stuff like this grabs the limelight, we hear less about things like how (only some?) states now ban married couples from sex if one is also suffering from dementia — because that partner can’t “choose” in real time.
I’d likely get aboard a “divorce government from marriage” bandwagon. If the government really should get out of the bedroom, don’t be all special-case about it. Just. Do. It.
At one time, I could claim that at least it got National Security more or less right. These days, US.gov is screwing up even that.