“Yeah—wasn’t the celibacy of priests a late (12th century or so) development in the Church?”
Yep. It was an administrative decision by the Pope. Parishoners were complaining that Priests weren’t doing their jobs because they were too busy working at other jobs to support wives and mistresses. So the Pope said no more marrying.
The Eastern churches ordain married men to the diocesan priesthood (but never permit them to remarry if they are widowed), but only elevate celibate men to the episcopacy.