On average, the game is very balanced. Folks just complain about individual issues with certain champs ("individual" as in: they cannot play around it and they assume that whatever playstyle THEY chose always has to work in all situations).
Further, people forget one thing (although, in all fairness, the ranking system encourages people to forget): LOL is and will always be a team game - champions are balanced (and have to be balanced) around their impact in TEAM situations.
In addition, the game has stages, and champions do well in different stages for each game. For (a super simple) example, if you get an all damage, all early team comp, and you lose against triple tank with Veigar and Tristana because the game goes to 50 minutes, that does NOT mean the game is "unbalanced". It means you did not play to your TEAM's victory condition.
Thinking that every individual player with whatever champ they "master" should be able to solo-carry themselves to Challenger divisions is WRONG. It is a myth spread by those who got lucky and got there, and people believe it and now call the game "unbalanced" because they feel they have mastered a champion, but still there are situations when they can do shit.
And yes, there are also different "metas" - playing Katarina all the time works well sometimes (assassin, carry, squishy meta), and it doesnt work well at other times (tank meta). That's how it is.