Whenever you introduce new rules into a system you have to consider all the negative consequences it could have. Imagine the worst possible thing that people could do with said system.
An example of a negative side to a vote-kick system is 4 people queued up together kicking anyone who doesn't play the champ they want them to. Or kicking just for fun. I'm sure there are many more.
Figure out a rule that helps with trolls that cannot be abused and you'll likely get more traction. And... good luck with that.
The essential problem is that for any system, a 'troll' is indistinguishable from someone who the majority of other players just don't like. Read about mob rule.
The other problem is there's no way to automate finding the difference between a 'troll' and someone who is not playing a strictly enforced meta. Riot does not want to enforce a meta, and I'm glad they don't. If they were willing to enforce a meta though, there are better solutions than a vote kick, such as limited champ/summoner selection based on what roles other people have already filled.