There are no "toxic people" and "non-toxic people"
##They're just people. People who become toxic for a reason.
Riot can't get rid of all the "toxic people" because "they" don't exist. You can see toxicity everywhere and in anyone, even on the Boards. Profiling people as "toxic" and "non-toxic" just shows a lack of empathy. Riot has increased anti-toxicity systems manyfold since the game's inception. To say that Riot does nothing or doesn't care about toxicity is a huge lie. Instant report feedback? Didn't have that in the past. Honor system incentivizing reform? Nope. A tiered punishment system? Nonexistant.
The older system--the tribunal-- was filled with false bans, little punishment (from lack of people going through and punishing toxicity in Tribunal and because there were no chat restrictions, so the system was hesitant on issuing bans), and inefficiency (not enough players to go through all reports).
Lyte (and WookieCookie & co., Lyte was just a figurehead) made a great attempt to get rid of toxicity in the game. In hindsight, I now think he saw that toxicity might wind up causing League to die. Many players feel the same way today. Many felt it earlier as well. When he implemented the system, he curbed toxicity, but only by creating the divide between "toxic" and "non-toxic" players. Toxic players would be locked with immense numbers of chat restrictions from the trigger-happy system. I personally probably got over a hundred games worth of chat restrictions. Many previously unpunished players, including videogamedunkey (where my current username came from) also got immense amounts of bans.
Riot has certainly demonstrated they care about toxicity in the game.
In that system, people could tell who was and who wasn't chat restricted, and those people--yes, those "non-toxic" not chat restricted people--would abuse the hell out of it. Sometimes it was just limited to taunting in the chat. Other times, they'd try to tilt you through gameplay, knowing you would be unable to vent.
Toxic people were sick; toxicity was treated like a disease, that needed to be diagnosed by the system, and then cured. "Reformed."
This ends up creating a kind of hierarchical class system between the two groups, and the same mentality is present today as it was then. There is literally a subreddit dedicated to Lyte smites..
If toxicity must be treated like a sickness to be "reformed", then let it not just be a disease that affects individuals, but something that has infected our entire playerbase.
Again, "toxic people" aren't simply toxic people. They're toxic for a reason. Obviously this reason may vary, but, from what I can see, the largest source is the drive to win and climb. I'm sure you probably see most toxicity mostly in losing teams. Again, Lyte & co. also knew this. They had ranked restrictions, with the intent to push hypercompetitive players into casual environments so they can focus less on winning (and will be exposed to other players who also think winning isn't everything). The intent of chat restrictions was that >"Limiting the number of messages a player can send turns chat into a limited resource. This messaging “ammunition” can either go toward useful comms that help coordinate a successful gank or toward flaming an underperforming teammate. Chat restrictions challenge players with a resource dilemma and a moral decision."
If toxicity is to be removed from the game, there needs to be less pushing for people to win and climb. I think over the years, greater emphasis on ranked and LCS(Look at the client menu) has fostered hypercompetitiveness, and, in turn toxicity. Players engage in toxic behavior because they care so much about winning and climbing. Too much. Make it so the game is less about winning or losing. Make loss forgiven systems to counteract and disincentivize intentional griefing (I think I have an idea that could somewhat work for this. But I wouldn't be the first to say that and be wrong). Get rid of the antiquated LP and promotion system, which focus on the value of immediate wins and losses. Something you see a lot of players telling people being toxic is that "it's just a game." Many people become toxic in the heat of the moment because they can't see that. Make them see that.
This isn't a problem just in League. Basically every remotely popular game is like that.