So why are the champions even fighting one another?
No, seriously.
If the Institute of War doesn't exist, why are all these random entities full out brawling with one another?
If Riot straight up doesn't want a lore that's fine, but they had better tell us if this is the case.
If Team Fortress 2, the most nonsensical madness shooter of all time can have some awesome lore and character development then I see no reason League can't. It isn't really a question of if the game is capable, it's a question of if Riot wants to pursue it.
So why are the champions fighting? Maybe they are trying to build a vacation house on the enemy team's nexus but property laws prevent them from doing so, therefore they take the next logical step... TEAM BASED COMBAT TO THE DEATH.
Edit: Also kinda want to add, moving forward lorewise, this is something Riot needs to establish ASAP. Without defining the basis for the entire League experience it will be difficult for lore to move forward in the future.
MUNDO FIGHT WHAT HE PLEASES