Put the League in the Future
Instead of removing the league, why not make it so it's a thing done in the future? Instead of having it so that the champions exist at the same time as the institute make it so the institute is a futuristic place where powerful mages have made constructs based off of legends, and they use those constructs to settle disputes, entertain people, blah blah blah. It's the league of "legends," right? Well, most "legends" are dead. It would also explain why champions who are supposed to be basic, like Garen, can do impossible things, like summon a giant sword of Demacia from the sky.
I know I'm not the only one who enjoyed having a reason why these people/creatures are killing each other, or enjoyed there being a reason why I'm controlling them, or enjoyed being a part of the game because of the story. When I was younger it amused me to imagine myself as living the life of a summoner. I've always held a little bit of that, and now it's been taken.
Edit: It occurred to me today that this would also solve the problem of why someone like Gnar is a champion. Champs like Gnar, Rammus, Orianna, they're just interesting stories- literally legends. What place do they have in the world other than to be interesting? Doing things in my way would give us a reason for why they exist. They exist because they're interesting legends who someone decided would be an interesting champion.