>>30058180I think the Pokemon League stipulates that official battles are 1 on 1, two on two, three on three or one of the other special rules, have a set of 6 designated battlers on you.
So while you could carry around a dozen pokeballs, trainers doing the rounds at Gym only carry their six battlers because that's all they can realistically focus on raising at once.
The games just don't deviate from the formula because the engine is pretty much limited to a few kinds of battles and for balance reasons.
