It's an interesting idea... But still the whole subjectivity is a problem.
Non-musicians don't know the first thing about piano playing, and they are the vast majority, so it wouldn't make much sense to let them decide. Then entertaining the audience would become a consideration, whoever plays Fur Elise and looks prettiest will win

.
Restricting it to a group of musicians, or maybe specifically pianists, would be a better idea. That would perhaps become more a reflection of the groups tastes, which may vary from area to area. It would be neat to try though.
Still, the more I think about it, the more I feel competitions are a silly idea to begin with. Like figure skating in the olympics, how can you objectively judge such a thing?