Another “wild” idea is giving militia-line the ability to take buildings (aka convert buildings) and remove this ability from monks, which always seemed an nonsense feature to me.
It is historically correct, and gives militia-line a solid role assaulting and taking enemy buildings.
How this could work? A raw idea is simply copy and paste the monks mechanic, just lowering the convertion rates a lot for balance reasons.
Of course, some buildings cannot be taked (converted), like TCs, Monasteries, Castles, walls and wonders.
Redemption still gives monks the ability to convert siege, maybe its cost could be reduced a little to compensate.