And no, one time in a normal during their free week isn't enough. I mean sit down and delve deep into them. 25 to 50 game minimum deep, and when you feel like you have some kind of mastery with them, take them into rank then where the stakes are higher, you can get properly counterpicked and people tend to know what they are doing and give more effort (tend to.)
I am not going to say that there aren't champions that aren't dumb right now, Vayne is dumb, Riven needs some toning, Hecarim is dumb, etc, HOWEVER, personal mastery of champions does FAR more then actual skills in kits and half these champions aren't anywhere near as easy and braindead as you think.
At times, this thought process will just confirm your suspicions, but more often then not, it will help you see WHY said champions have the powers they do and even teach you ways to beat them. I honestly believe champion QQ would plummet if people actually did this.