As I understand it, Eastern Catholic churches do allow married priests, but under THREE conditions:
One: That a man be already married BEFORE studying and being ordained.
Two: That if his marriage is annulled while a priest, he is forbidden to remarry,.
Three: The highest a married man can go is to be a priest. Any higher, he is to be unmarried.
Substantially correct. A man must marry before ordination to the diaconate, not necessarily before entering seminary. A cleric cannot remarry, even if his wife should die. And Eastern bishops are chosen from among monks, who are always celibate. An ordinary diocesan priest isn’t eligible, unless he was ordained as a celibate, or widowed, etc.