I have some problems with this but I'll try not to tread too much on what Neverworld said, as he does make a good amount of points:
1.) Who decides what's best? We have some designers who go above and beyond with numbers and balance but are as bare bones as it can be when it comes to characterization. Others, such as myself, go above and beyond when it comes to characterization but fall flat when they have to balance and attach numbers to a kit. So in reality there'd have to be several contests for...
Doesn't sound like much but that's four categories. Leave one of those categories out and you're left with those who are vastly upset that their category meant nothing. ("Omg that character was well balanced but the lore was garbage" or "Wow, way to pick the guy with the worst kit. Who cares about lore?" )
2.) Let me start by talking about
. Yasuo was interesting because from his conception to full realization, his entire kit was gutted. 100% nothing of his original kit remains. Why am I saying this? Because as a Champion is given life and built, things change. Sometimes it's not feasible to make this champion with the engine available. Other times it's just not working so they scrap a skill. The point here is that things change and, often, a champion doesn't much resemble their original creation. At what point will the creator feel like their original idea is lost?
3.) Champion design takes months/years. Sure, the news of a winner would flare up but quickly die down. "Oh just rush the concept ASAP" the pipeline doesn't work like that. In fact, several Rioters have said it's less like a pipeline and more like a massive maze that takes a while to clear. It's entirely possible this concept might take ages to come out into existence. While the winner would remember, it's very likely that most of the community will have forgotten.
4.) Time. Time time time. The contest itself would take forever to run. The final judgement would take even longer. And the creation-to-release would go even further. So in the end you have a massive contest that takes up the majority of Riot's year while also requiring the attention of several champion designers. Not exactly feasible for a company like Riot who has to continually pump out content and balance changes.
5.) Original Creator Rights? At what point will the original creator stop having a say in the creation? Will Riot have to hire them on to work with this? Etc.
As much fun as it'd be, it's just not feasible to do.