In all honesty, none of these points have anything to do with why we used wireless controllers. The lag is so negligable that it is not an issue. It's the player's fault if he doesn't replace the battery, ect.
The reason that we use wired controllers is because wireless controllers are banned at large scale tournaments. For wavebirds, they are banned due to interference issues, considering the larger the tournament, the more likely you are to find someone using the same frequency.
Wiimotes are banned because of the time consuming proccess of syncing them everytime you move to a different system. With so many Wiis, you'd often sync to the wrong system and have to sync again. This can become a time consuming process, and is gotten rid of by simply banning the use of wiimotes. Having people leave two wiimotes at each system brings up the problem of who is responsible for the controllers. They can be broken while you're playing a match, stolen, ect. To solve that problem, we have people bringing their own controllers and keeping them with them at all times.
The only viable solution to all these problems is to only allow wired controllers. Unfourtunately for some, the game cube controller is the only wired controller, and thus the only choice at large scale tournaments at the moment.