As soon as you mentioned CONTRACT you illustrated why the libertarian model of "private" marriage has never existed, and never will exist--and hence is nonsense. "Private" marriage is not just a legal fiction, it is fiction period.
Marriage, by it's definition is a publicly & legally acknowledged relationship, involving a covenant or contract, between a man and a woman....and actually the community.
WHAT, THE COMMUNITY? ARE YOU CRAZY? you say?
Yes, the community--in in that contract shared property and (usually) children are involved, hence you have issues of law:
1)If and when one spouse dies (inheritance & adoption issues)
2)If or when one spouse leaves/divorces or disappears
3)Regarding benefits or privileges allowed by the government (there are many (many).
Because of those 3 realities the State has ALWAYS recognized marriage--as a legal state of life--even back before marriage licenses--where the Church blessing was all it took to make it legally recognized. The abolition of marriage licenses may of been possible before the complex of government benefits/priveleges of the modern welfare state--but until (or unless) the welfare state is gone--(#3) there can be no doing away with marriage licenses.
Even without a welfare state (news flash: it ain't going away any time soon!)a Church form or some other legally binding format would have to exist....becoming the de facto "marriage license."
Even such things as mortgages and insurance rates are varied--depending on marital status--and only law....can determine who is really married, and who is not. Once law in involved, the State is involved.
Since marriage and the family is THE primary building-block of civilization and government...it does no good to naively call for a "privitization" of marriage--as by definition, marriage never has been, and never will be private.
I'll give you an example: Two lesbians "marry" and adopt a child (wait, do we allow lesbians to adopt? Do we allow singles to adopt? What's single, what's married, with "government out of the marriage business?" They separate, and both want the child. But were they "married" in the first place (I would say under God, not...) A friend of their's with a mail order "ordination" in the Church of What's Goin Down had a little ceremony for them--and s/he has disappeared in San Francisco's transgendered community. They have a piece of paper signed with an illegible signature by the guy/girl they say has disappeared...from a "church" that never really existed either. Who gets the kid? Were they legally married or not? Who get's the Miata? The hot tub? How would any such set of conundrums be resolved in the mythical libertarian world?
Both by default, changeable by mutual agreement.
Who get's the Miata? The hot tub?
Whomever the private marital contract said got those things, instead of Big Government courts deciding who got those things.