I actually thought about this as well, Grahr. If we were locked into one meta, then this would be great. Unfortunately Arcticfury is right, and it's not possible.
Something to branch off of that, though, and keep in mind I'm just throwing this around for the sake of brainstorming..
What if there was a menu where you could post what you wanted to play, how you plan to play it, and where you would like to play it, and you could send invites to random people (you wouldn't know names or stats until after the game you're about to play is over) and fill teams that way? It would still be random people, you would just have a somewhat shaped meta going in.
El Faty, while I agree with your statement, it's not just the fact that YOU have to know every lane/position going into a game, but the rest of your teammates do as well. That's what is frustrating. While I could cover any position, you could have 3 people locked, mid/top/adc, you take support or jungle, but the 5th guy can't do either of the last two. That's where I keep getting screwed over. Being able to hand pick people by where they want to play could possibly help solve some of those.