Many states got rid of the concept of a common law marriage (which in states that still recognize it is proved by the couple's living together and holding themselves out as husband and wife) because of problems of proof. Mere cohabitation was never enough, nor was having kids. Those states preferred something more objective, like having the couple obtain a marriage license that is signed by someone authorized to conduct a civil marriage and that is filed of record. That way you don't need to find witnesses.
In any event, the government (via its court system) will still be "in the business of marriage" if it is ever called upon to determine who someone's spouse is.
If the fact that government courts might sometimes need to resolve the question of whether someone is or was "married" in order to dispose of some particular matter is sufficient to establish that government is (or should be) "in the business of marriage", then there is literally nothing that government is not (or should not be) "in the business" of, since a definition of any given thing might sometimes need to be addressed in a government court in order to dispose of some particular matter. Thus, if the aforementioned fact warrants the claim that the government is "in the business of marriage", then that claim is capable of meaning pretty much anything at all (or it is essentially meaningless, which amounts to the same thing).
Most marriages have always been in some way explicit, as well as documented and/or witnessed, etc., according to customary sensibilities, traditions, etc.
[1] (and this does not for that reason make them any less "objective" than politician-defined marriages). They are perfectly capable of being and remaining so without any need for politicians and bureaucrats to usurp the issue by arrogating to themselves sole (or even any) authority to define or "regularize" the issue for the sake of their own convenience (and power)
[2].
If there are those who (due to a lack of such explicitness, documentation, witnesses, etc.) find themselves unable to adequately establish their status as "spouses" to the satisfaction of some court, then that is their problem, not the court's (or, by extension, the government's). Let such people stand as a cautionary example for others, and as an incentive to avoid such problems in case of later disputes (that is, after all, exactly why marriage customs involving documentation, witnessing, etc. came to be adopted in the first place - not because some politicians thought it was a clever new idea no one had come up with before).
[1] Such customs, traditions, etc. may (and almost certainly will) vary from time, place and/or culture to time, place and/or culture. For example, San Franciscans might well have different standards for what constitutes (or suffices to establish) a marriage than Peorians do - and that is just as it ought to be. Let the cosmopolitan be cosmopolitan and let the provincial be provincial. There are virtues and faults to be found in either, and outside of initiations of force or acts of aggression
[3], neither should be permitted to dictate to the other what it must value (or disvalue).
[2] For example, I am sure it is quite convenient to the state for there to be some state-appointed (or state-approved) persons who are exclusively "authorized to conduct a civil marriage" to the exclusion of all others (especially in order for the state to collect, compile, and control information about private citizens - or to wield greater power over whom is permitted to marry whom, as in the case of anti-miscegenation or anti-gay-marriage policies). I'm just not seeing why it is necessary (as distinct from convenient) in order to be able to adequately establish whether someone is or was married (which is quite able to be done without the involvement of superfluous, specially-authorized bureaucrats).
[3] And even in the matter of "initiations of force or acts of aggression", there will at the very least still be cultural and jurisprudential differences on the margins (e.g., over what are and are not "fighting words"). Libertarians
qua libertarians do not and can not have a "theory of everything", and neither do or can the feds (try as they might to the contrary) - and that includes things like "marriage".
