The only public record of your marriage is "X and Y got married." If your marriage included a prenup, that information is kept private. You don't have to state for the public record precisely why you feel like getting married. Seems fair the only public record of getting unmarried should be "X and Y got divorced."

A marriage is a private contract between consenting parties just like any other. The state should not get involved.

Maybe that is how it should be, but that isn't how it is right now.

If you don't want public recognition of your marriage, I'm sure you could write up some private contracts, but it won't confer all of the same rights that a state-recognized marriage gets.