Who did so before the State took over licensing marriages and then allowing divorces?
Marriage has been going on for thousands of years. Christian marriage has been going on since Christ instituted it as a sacrament.
Up until about 150 years ago (and more recently in most places), the maximum sole function of the State (at least in Christian days) was to document what the Church had declared sacred. And in many places, the State even stayed farther away than that. Admittedly, in Northern Europe and England, where Protestants declared that marriage was not a sacrament, the State's involvement came about a lot earlier than in the Southern, more Catholic, parts of Europe. But the idea of divorce didn't really start to gain ground except as a consequence of the Enlightenment...where the influence of religion started to be minimized. Again, this hit Northern Europe earlier than Southern Europe (that's probably why Leo's encyclical, quoted above, wasn't written until 1880).
Face it, with the exception of Henry VIII, divorce didn't hardly ever happen, either in Protestant or Catholic Europe. But if separations happened, how were they handled before the State courts got involved?
But you’d take away a lucrative Dem-laden industry of “family” court and divorce lawyers and that would make rats sad.