What you saw was marriage bands, published and filed with the state announcing the marriage of so and so. Licensure was never required for marriage in a church as a religious rite until the 20th century.
sorry, banns, not bands.
Louis, state laws differ. You may be correct in some states. Perhaps in Virginia or other states which had a tradition of a state-supported Anglican clergy until the late 1700s, some vestiges of that may have remained into the 1800s with marriage laws.
I am quite aware that under frontier conditions, circuit-riding ministers had to keep a record of marriages since the closest county official could be an impossibly long journey not just of days or weeks but of months. That was the exception, not the norm, and became less and less common as counties began to be organized on the frontier, with the goal of having everyone live within a one-day horseback ride of a county courthouse.
That certainly was not what I saw with marriage licenses in the old archives of county courthouses in Iowa, or for that matter, genealogical records of marriages in other northern states. I have done very little genealogical research in the South so I can't speak with any level of assurance. I did have a few Quaker ancestors on my mother's side who moved out of the South prior to the Civil War and then left the Society of Friends, but I wouldn't want to extrapolate from those ancestors to other churches. I'm prepared to believe that the marriage practices of the Society of Friends might have been quite different from those of other churches in the same states in the same time period, and I don't have enough information to know for sure.
You may or may not be aware that outside the Anglican and Roman Catholic traditions, marriage for Protestants has historically been regarded as a civil matter and church weddings were prayer services for the couple. In some Protestant nations, two separate marriage ceremonies were conducted in reaction against a Roman Catholic view of marriage as a sacrament. It would take me more time than I have today to track down and post links to documented proof of the marriage practices of Puritan New England, but there's no question that by the early 1800s, marriage licenses were being issued by and regulated by the civil authority in states which were populated largely by people who had formerly lived in New England.
The bottom line is that your view of marriage not having state authority behind it until the early 20th century simply cannot be documented in the historical record, even in the United States. You may be right about Roman Catholic countries where the church took over numerous functions during the Middle Ages due to the collapse of the civil state and a sacramental view of marriage, but that model would be hard to find outside of Europe and nations composed largely of European immigrants.