I don't mind stat blocks in a quest-centric roleplay or (obviously) in tabletop gaming, but I don't really think they work well for something like a fighting league or a tournament. Stat blocks can't exist in a void very easily; they have to exist within a system of reference like a tabletop gaming system, which the stat block interacts with. These systems are often limited, and make it difficult to represent the broad variety of characters you see people play in free-form where "setting" is not a limiting factor. Finding - or developing - a system to cover the gamut of characters that people play has always been an enormous hurdle to using stat blocks.

It's not impossible, but you would almost certainly have to narrow things down a great deal. The narrower the scope, the narrower the appeal. If you settle on x or y system or x or y setting, then you're cutting out a, b, and c. It's doable, but you might have to be willing to accept casualties in the form of a smaller player base. Even if you narrow it down to "fantasy + mild powers", it would probably still require more specificity, especially in regards to magic or supernatural abilities.

Which isn't to say that you shouldn't do it, or that it's not possible.. just that it's probably going to be a lot of work. It's also not a guarantee of things being any fairer (or fair at all), because within any system, there's always a means to game it for an advantage. 

(As an aside, I'm all for internal stat systems for characters. I just don't expect all characters - or even two characters made by the same player - to adhere to the same system, because people almost always develop the system in tandem with the character.)