The idea of unmarried clergy is pretty much only in Roman Catholicism and even that is relatively modern. Now, in the Middle Ages there was a religious order of sorts called Beguines and Beghards. They lived as monks and nuns except they could marry and retain property. And that was repressed in the Late Middle Ages.
Yeah, the Roman Curia was generally quite unhappy with them all along, although not always actively persecuting. However, they were not technically
priests.
The blanket ban on marriage for ordained priests exists in principle, IIRC, from mid-eleventh century onwards. It is important to note that it was never a matter of
doctrine strictly speaking (although it is justified in reference to Scripture) but of
discipline alone (therefore exceptions are allowed on grounds of tradition for Eastern Churches of Roman obedience).
The ban was rarely enforced prior to the Fourth Lateran Council (1215) which reiterated and strengthened it, timidly trying something approaching enforcement. However, until the Council of Trent, almost three centuries and half later, it was commonplace for priests, especially in rural areas, to have "marriages", as in, stable female partners who lived with them, with the flock being aware, (and quite happy, often) about it. It was not recognized though.
All along, however, and from relatively early on,
bishops could not marry (which is also the case for Orthodox Churches; therefore, bishops tended to be monks, who also, of course, were bound to celibacy roughly from the start)
Now, the Catholic blanket ban is divisive, setting Rome apart from both the East and the Reformed Churches (and it is also controversial among Catholics in recent times). So priests allowed to marry are one less reason of division within Christianity. On the other hand, this means that the Pope has overall less control.
If
bishops are allowed to marry, on the other hand, you'd quickly see dynasties in Western Europe. There would be a lot less separation of the temporal and spiritual realms, with many bishoprics turning into dynastic petty kingdoms, and kings being ordained as priests. Endgame is local churches with only modest, if any, central control. Recall that, while the Roman Church was a meritocracy in principle even in the Middle Ages, the top brass was almost invariably recruited among nobility. The "meritocratic" aspect would probably survive very poorly if at all in context where priests an marry and their sons can inherit.