Not necesarily. I'm a real estate lawyer, and most real estate contracts stipulate that any disputes will be handled by a private arbitration organization- government courts never get involved. A lot of contracts are handled that way these days in many fields. However, your point is essentially correct- any private contract requires a mechanism whereby a third party can resolve contract disputes.
It is possible in this country for sectarian institutions to continue independently of the state administered institutions.
Do you mean that, for example, a secular divorce has no effect on the status of a marriage within the Jewish or Catholic faiths? There are rabbinical courts in Brooklyn where Orthodox Jews go to get their marriages dissolved under the requirements of their religion. Such a rabbinical ruling, however, in no way affects the validity of their secular marriage, and vice versa.
For this reason, I believe we are talking about at least two forms of marriage that are at core unrelated although superficially they appear similar.
If we are talking about the same thing, I agree completely. There is the secular institution of marriage and also the religious institution. However, either one can exist independently of the other.