I hear a lot of people talking about kill quotes in game (i.e. if Diana kills Leona, Diana could say something like "The sun sets"), but that sort of thing would probably require a lot of work to put in place. They're already trying to bring more lore into the game with the Shadow Isles map, and I'm sure Thresh will have his own shrine quotes on that map. So I feel like Riot must agree with you; they're trying to put lore in the game so that you experience it by playing. Maybe soon they're unveil whatever it is they say they've been working on - it might be something like the game mode you're describing.