well, 1v1 cover up a part from the game, but not the entire thing.
only thing u can proof in a 1v1 is how good your laning phase is till mid game. but technically that doesnt covers up how good you are in mid/late game (roaming, teamfights, dragon/baron control.)
as example, i would loose every 1v1 because my laning phase sucks hard. but i would own/carry hard every mid-late game, because im there much stronger in terms of play.
sure i could pick a strong laner and still win (ie, zed, renek, khazix or something.) but it still would not proof how good the rest of the game would go. (u could get fed up on other lanes, by roaming, u can get drake to make your gold even, etc. etc.)
if u want to proof who is better get everyone a 5v5 team and make a match, whoever wins afterwards is the better player imo.