While I agree with you that marriage is an implied (or actual) contract, I don’t see any reason we can’t still have that. Maybe have domestic partnership agreements drawn up in lieu of a marriage license? As far as children go, they already belong to whomever is listed on the birth certificate, regardless of parents’ marital status.
That must be why we never have custody battles? By the way, you don't need a marriage license to get legally married in the United States.