I've noticed that nearly every champion rotation, the forum chooses a new favourite punching bag; a champion that counters someone's main so well that they complain on the forums. Rengar was the punching bag of last rotation, Talon is becoming the new one this rotation, and Renekton comes up in conversation so much more often.
Imbalances aren't a huge problem in games, but in sports, they are. Arguably, mild imbalances create fun in a game. League is a game wrapped up and marketed like a sport, so players often adopt a sportsman mentality of 'the game should be fair', rather than a gamer's mentality of 'what are the exploits/how do I win.'
In my previous MMO, World of Tanks, your choice of tank influences winrate by up to 31%!!! That's absolutely insane, and part of the reason I left the game. I also left because, unlike League, rewards are zero-sum. You get rewarded in World of Tanks for taking damage, and thus rewards, away from your team mates. League is despite imbalances a well-designed game; objectives, individual skill, and teamwork play a bigger role than the champion you choose. In my experience, the champion you play in League has maybe a 10% bearing on your winrate.
I'm sure you've also seen diamond smurfs go 35/4 and get all 5 dragons, so in anecdote at least, you can observe the power of skill and teamwork. I've also had the pleasure of playing against many newbie Zeds and Volibears, and seeing diamond players on the same champs in solo queue (not against other pros... because I'm a bit terrible T.T)
Zed and Volibear are underwhelming when the player doesn't have practice, yet can carry games alone with a good player behind them.
Officially, I think some champs would be less hated turning numbers down a notch or two, but they aren't outright unfair.
TL;DR
Champions don't inflate winrate appreciably. Players who can exploit the game do inflate winrate, regardless of who they choose to play.
EDIT: The estimate of 10% proves accurate, looking at the variance in average winrates by champion; 42% is the lowest, 56% is the highest, so champion selection averages about 12% into your winrate. Keep in mind that although skill and teamwork are averaged out of that 12%, they are still present to some small degree due to error. For any player, regardless of skill, champions in League can give you up to a 12% boosted winrate. In World of Tanks, your choice of tanks a whopping and clearly unfair 31% of your winrate.