You can tell me to get rid of this answer if you want, I don't mind.
#My Illogical Theory
(I haven't done one of these in so long omg)
I'm certainly sure the Pokemon you call upon with the Ride Pager do not belong to the Trial Captains of the islands. I say this not only because of the points you listed above, but also because most of the Trial Captains and the Elite Four do not battle with anything that could be a Ride Pokemon; Tauros, Stoutland, Lapras, Sharpedo and Charizard are species that do not belong to any important characters in the storyline (this excludes Trainer Red at the Battle Tree and Youngster Tristan, a trainer who attempts to dethrone your Champion when you rematch your Pokemon League). Although, this can be contradicted by the idea that the Trial Captains or Elite Four wouldn't need to necessarily battle with those Pokemon.
There's also a quote in the game that specifically states that a registered Ride Pokemon would come to you when you call them. The quote was written to indicate that there existed many Ride Pokemon of the same species... I'm sorry I couldn't find it. If anyone is able to find it, please comment below!
Hope I helped. :)
Source: Respective Pokemon species' TrainerDex entries on Serebii.