I'm currently working so I can't be as in-depth as I'd like; however, there's interesting discussion here so I'm going to throw in some additional details to clear up some misconceptions.
1) The average Tribunal user actually has really low toxicity compared to the rest of the population. Whether you are looking at Offensive Language, Negative Attitude or Verbal Abuse reports or simply numbers of reports, the average Tribunal user has lower averages than a random sample in the population and way lower than the average toxic player. Saying the "trolls" or "kids" are the ones voting in the Tribunal is just false.
2) Over 50% of players who get a warning from the Tribunal understand what they did wrong and reduce their toxicity dramatically--these players never come back to the Tribunal again.
3) That % that reforms gets lower and lower with multiple punishments in the Tribunal, suggesting that if a player doesn't 'get it' within 1-2 punishments... they are unlikely to ever understand why their behavior is inappropriate in League of Legends.
4) A large percentage of players actually never harass, flame, or use racist/bigotry/abusive language towards other players. Surprisingly, the average player in League of Legends doesn't use this kind of language so saying "it's the internet and everyone is bad" just isn't true.
5) A large number of players actually do not know what the Tribunal is--they don't visit the forums or the website and are not banned by the Tribunal... so it's an unknown entity to them.
I know the Tribunal isn't perfect; but, it does what it's designed to do well which is to identify the most toxic players in the game and either reform or remove them. Should we be designing other systems for the good players and the other 98% of the population that may never interact with the Tribunal? Yes. We know the Tribunal by itself isn't enough to tackle player behavior issues in the game; but, it's one piece of the overall solution.
First, saturation is a major issue. As a player between 900 and 1000 elo, I can say with confidence that there is at least one person per game who does something bannable by tribunal (AFK, verbal abuse, intentional feeding), and I'd be willing to keep a record to prove it. You can point to statistics that say 98% of the whole population of players never get into tribunal, when in reality it *feels* like at least 10% of players deserve it (statistically speaking on a per-game basis). It's a nice statistic you've found to be able to point to, but that statistic alone presents several problems. Is this active accounts only? Or does it include accounts that were made and then abandoned? That's just inflating the population with a demographic that isn't a factor. Abandoned accounts don't join a game, threaten "mid or afk", then proceed to rage, intentionally feed, then leave a game.
Second, Tribunal's effects aren't measurable when banned people can just proceed to make another account. I've noticed when leveling up a secondary account that there was an overwhelming number of trolls/feeders in the first 10 or so levels - I'd seen as many as 4 in a game (I'm not adding people who don't know/are trying to learn the game mind you - this is buying boots of mobility and charging into their tower like no tomorrow).
Third - the number of people I've seen ignore/refuse to report a person clearly in violation of the summoner's code is staggering. The majority of them come from the opposing team to the offender - if team 1 has an intentional feeder giving team 2 a free win, I've seen many times where team 2 just laughs and says they won't report since they got a free win and don't care. If someone from team 1 goes afk, team 2 will laugh and say it didn't matter and that they were going to win anyway. While many people do report and have a sense of justice, just as many people are supportive or that behavior.