What are the societal implication of a feudal system where a few wizards take the place of entire armies?

I’m working on a game about wizards fighting wizards. I’m trying to world-build, and would love people’s input on what am missing, or not thinking through.

The game takes place in a middle age fantasy setting. About 300-400 years ago, there were a couple of key discoveries of magic that transformed it from little tricks to a weapon of warfare. Within about 50 to 100 years of that time, magic had replaced armies (I’m imagining there being a turning-point battle like Crécy or Agincourt, except instead of a battalion of archers, it was a few wizards). At that point, wizards took over as the rulers, like medieval knights and kings. It began as one big empire of wizards, but within 2 generations that fell apart into small feudal kingdoms. Nearly all of what’s been learned of magic since then pertains to fighting or warfare. As such, wizards still rely on peasants to pay the taxes that enrich them (growing food, mining gold, etc). The wizards are territorial about the regions/countries they control, and protect the people in it (somewhat, they still employ town guards for basic policing). Magic is based on knowledge and training, so while some people have a natural knack for it, it does need to be taught.

I’m trying to think of what the big societal implications would be. Here are a few things I’ve thought of:

• Most wizards would train a few apprentices, basically as protection. Each apprentice would be assigned different parts of the kingdom (like lesser nobles). The apprentices would do much of the fighting. The top wizard would stay on top though a combination of being the most deadly, and though loyalty; their deaths often lead to fights about succession.
• Wars between wizards would mostly be caused by greed (wanting another’s land) or fear (wanting to protect themself from other wizards). I'm debating how common this would be.
• In many kingdoms, magic would be passed through blood-lines (e.g. wizards would train only their children to be wizards). But, I guess, would it be possible that in some kingdoms, wizards would seek out promising peasants (as they'd make the best apprentices)?
• If a wizard’s power was based on their knowledge, reflexes, and intuition, then women could be wizards just as easily as men. I’m wondering what the implications of wizard-knight-queens would be both for the structure of royalty, and greater societal effects.
• Magic is a well-kept secret. Wizards try not to train too many apprentices, because they’re worried about being dethroned. That said, secrets will leak, and sometimes peasants or people on the outskirts will learn little things.
• Kingdoms will differ, but in many, the wizards are worshiped as gods.

I guess, what am I missing? Does any of this not make sense? What else should I think through? Thank you!