Did bishops(or any other member of church, excluding holy orders ofc.) really lead armies and took part in battles in medieval times?
the game is an abstraction. the current mechanics do not cover everything. whether or not battle bishops existed in reality is in the same vein for women leading armies in medieval times.
we might as well ask if kings allowed lowborns to lead armies in medieval times,
and if kings could be crusading in the levant one day, suffer a defeat and suddenly be defending his castle in northern europe against pagans the other.
or more relevant to your question, if Popes really lead armies with themselves at the forefront.
the game is an abstraction. the current mechanics do not cover everything. whether or not battle bishops existed in reality is in the same vein for women leading armies in medieval times.
And also at the Battle of Hastings. Odo, bishop of Bayeux fight there. And he became the Earl of Kent as thank. With picture!
Another example is that two Hungarian archbishops commanded soldiers and died in the Battle of Mohi against the Golden Horde. So bishops leading soldiers is historical and not that rare