It’s always seemed to me that Aryon wasn’t born into a “Telvanni family,” and that he instead joined the Great House in early adulthood to pursue his own dreams and ambitions. There are many reasons I think this, not least his portrayal in House Telvanni as a precocious upstart with something to prove (and the classic “new money vs old money” dynamic between him and some of the older more established councilors). But it also seems to me that Aryon was raised in the Dunmeri faith, which I think would be unusual if his parents were Telvanni. Maybe not that unusual if they were just smallfolk or peasants on Telvanni land. And it would certainly not be unusual (and indeed almost a given) for any non-Telvanni Dunmer born in Morrowind.
Aryon is not by any means a pious or even religious man when we meet him in TES III. But he lacks the completely active and blasphemous distain for faith that the other Telvanni have. He seems to at least believe the Temple is useful, and he even built a temple for Tribunal priests to serve his smallfolk in Vos. Most Telvanni do not see any such use for or care about the Temple. It could be that he remembers the usefulness of the priests having served his community in his youth.
When you are gathering the votes of the Telvanni councilors to name you Hortator, whether you are his protege or not he hardly needs any convincing when you describe to him all the portents and strange occurrences of your story, and he commits to give you his support on the spot. It is possible that he merely sees this as an opportunity to get you to remove Gothren for him, of course, and he is relatively upfront about what he stands to gain from that. But to me, he also seems genuinely convinced, like he’s giving weight to the portents.
Later, after you inform Aryon that you’ve gathered all the votes, he tells you “I wish you luck, Hortator. I may even pray for you.” Let me say, as somebody who was personally raised atheist by atheist parents, it would literally never even occur to me to tell somebody that I would pray for them. I wouldn’t not do it out of spite or anything, I just wouldn’t even think of that. I don’t think he has been an active or practicing member of the Tribunal Temple for many many years, and he does imply that it would be a big deal for him to pray for anybody. But this still leaves me with the impression that he had some sort of spiritual training in his youth. I even believe that he does end up saying a prayer for the Hortator of House Telvanni, though maybe in a rushed sort of mumbled way. He hadn’t considered himself spiritual for a long time, but with all the weird things culminating on the island and this strange Outlander seemingly touched by destiny – well it couldn’t hurt.