With the writs I imagine the "some reason" is because it justFor some reason, all the Writ of Challenge movesets for at least gym leaders and elite 4 members (I dunno about regular trainers, I dunno if there are regular trainers, I've never played XY) are just level-up moves. Same with Maxie. Maxie also had level-up moves in the original, but I dunno, I feel like that's not something worth referencing in a remake, if indeed it was an intentional reference.
okay im about to speak out my ass on this but
It's just running a leveling routine on them, and that routine has it "level up" the movesets rather than retain their current movesets and they did this rather than insert another 3 or 4 or whatever variants for all the leaders & such at different levels.
The Battle Chataeu does indeed have normal trainers in it. The gym leaders/e4/champion only show up inside once you hit certain thresholds (like i think you have to be certain rank and done stuff like beat the game). normal trainers do not, so far as i know, have special movesets and just use whatever defaults.