You can’t privatize marriage because government will always want to know, and often be dragged into the decision of, who gets the property when the marriage ends (and thanks to death they ALL end at some point). That’s the reason government is in the marriage business in the first place, and you can’t get it out.
Millions of people shacking up would appear to be proving you wrong.
The reason government got into marriage in the first place was to regulate miscegenation.