Then, perhaps once a blacksmith is built, militia get to use iron-based equipment, for example?
Yes, I think that would be a good idea.
Blacksmith -> Metal Weapons and Armor
Armorsmith -> Masterwork Armor
Weaponssmith -> Masterwork Weapons
(or some rough equivalent)
However, legendary equipment and magical equipment ... I am less sure about.
Having lightning hammers on peasants (for no good reason) should certainly be an undesired anomaly I think.
Hmm.
Perhaps, if you go along the magical path instead of the warfare path ...
certain magical buildings would equip your militia with magically enchanted weapons rather than iron/steel weapons.
(probably only for Tower/magic specialized cities though ...)