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I mentioned this in another thread, but what if you have to "learn" the pose for each different super move during your journey? Like, throughout the story you encounter a Pokemon, your character has a moment of profound enlightenment, then you gain a deep enough understanding of the type to unleash the super move for all Pokemon of that type in the future.
Corocoro mentioned something about the starters and Rockruff, so those four might be the triggers for their respective type. Maybe there's a forced encounter with Rockruff early on that demonstrates this mechanic, similar to the elemental fang Poochyena in ORAS.
Adding to this, you and your rival encounter Tepu Koko at the very start of the game. Maybe the encounter is where you learn the Electric move? That would explain why the rival gets a Pichu so early in the game: to demonstrate the new electric beam attack (would also explain why the route 1 bug evolves into an electric type, so that you can make use of your move as soon as it evolves). Maybe collecting these is one of the goals of the game? If gyms are still around, maybe you also get them after battling a gym leader, because you've come to understand their type? It would also explain why you're seeking out the island guardians.