Actually that is a very good idea, hacking a PSX pad and adapting it to NGC. That would eliminate the analog control problem.
I don't understand why you want a gamecube cab though. Gamecube is my favorite current generation system, but I don't think its games would fit in a cab well. Then, getting rid of the analog control is going to limit your game selection even more.
I think most people use the xbox because they hack them with a mod chip, so they can play MAME.