Imo if magic existed, I would be scared and kinda want it to stop too. Like yeah if you can wield that magic it's cool, but if you can't... Not everyone is as lucky as Zoe, getting chosen as an aspect. Not everyone get's a master that teaches them wind magic sword stuff like Yasuo. So when you're one of these people and you come face to face with something as horrifying as Karthus, Evelyn or even just hearing the tales of rune wars, you wouldn't be scared? I would definitley be on the side that wants to stop magic, even if I had magical powers. Like the magic on runeterra has really shown just how catastrophic it can be and it's horrifying. You could be mind controlled, you could be tortured to death, you could be tortured even after you're dead, etc etc. Like I understand non magic living has it's horrors too, like just murders with knives, but those are easier to control than a murderer that can send 10000 floating knives at you. Tell me that if you heard of Irelia you wouldn't be spooked as hell. She can slice you in half in the blink of an eye and there goes your life. You no longer get to go home to your family, no longer get to watch your favorite show, no longer get to eat or read books. You're just dead and it wasn't even close to a fair fight. And as your decapitated head is on the ground, still allowing you to view what's happening, you see everyone near you also getting murdered in the blink of an eye by magic flying swords. Your last moments on runetera were filled with horror, and who even knows what's going to happen to your soul. It could become part of the harrowing or Thresh's play things. Tortured forever and ever, no stop. Magic pretty scurry if u ask me I wud stand behind a big strong anti magic (but secretly kinda magical) man like garen. Even healing magic has it's horrors.
TL;DR: magic is scary and I would hate it too, so I don't think that the leaders of Demacia just hating magic would be that bad.