Well, there are quite a few things that bug me about the bans.
First, given Riot profits from players creating new accounts and dumping money into rebuying champions and boosts to get back to level 30, they could possibly have an interrest in nuking accounts knowing the player will have to pay them to relevel or take eons.
Second, due to it being free, it is a cesspool of kids under 18. Most people act the age of the group they are with. IE If you put a 13 year old with adults, he will act adult, and vise versa. This creates a very very negative player experience in randoms in regards to maturity.
Thirdly, it is a private game and as such you are governed by Riot and Riot alone. This basically means that since it is their game, you should not expect fairness, justice or being able to explain anything. The bottom line for this is they have to keep costs low and really don't have time to listen. They try to keep everything as statistically based as possible and simply come up with a mathamatical model to flag out behavior they don't like.
Fourth, connection testing. I became interrested in this issue after having been recently banned for connection issues. I had dropped 3-4 games after moving and had Clearwire 4g. I had tried to troubleshot the issue with their techs and playing another 3-4 games of bot games. Had no idea people could even be banned for dropping a bot game. Is it fair, meh, does it matter if its fair? No.
So what is the Tribunal, why was it setup the way it was?
It is purely a method to train the LoL community to do Riot's work. If they can get 5 people with a 99% chance to give the same punish as a riot employee would, then if all 5 of those people vote punish, you can say with almost certainty, he should be punished. This is why it is setup for you to make the same decision as Riot even though we all have very differing views of right and wrong. They will find all the people with over say a 75% success rate and take only their decisions into account. They will then see if there is a very popular consensus. If say over 75% of those make a vote one way or the other, the Riot employee could simply not review any information and hit the punish button. SOOO much easier than reading stuff.
If I had to do the system myself, I would do it very similar but with the player given an opportunity to explain the actions. The Tribunal would then be given a chance to pardon or not. If pardoned, Riot would never see it. If he wasn't pardoned, the ball would still be in Riot's court. This would allow the player the ability to explain himself and the Tribunal to save or not save. I also think the game should require some modest payment if they ever have any hopes of a community beyond vomit spewing teenagers.