The Order of Arkay performs burial rites and such. The Temple of Stendarr has various charity causes. The Resolution of Zen might be angel investors. What exactly does the Akatosh Chantry do or stand for? That priest who wrote The Akatosh/Alduin Dichotomy considers Akatosh and Auriel to be much the same as do some High Elves he interviewed and both Wayrest Bretons and Falmer have chantries, so I assume they do similar things, which involve singing and seeking "enlightenment". Details?

Perhaps learning about the nature of time gives worshipers insights into the truth of the past and the possibilities of the future? Of course, the Elder Scrolls already fulfill that niche. Perhaps it depends on the aspect of Aka being worshiped, as well. I've seen it put forward on this forum that Auriel looks back at the past, Alduin moves things forward into the future, and Akatosh deals with the present... or something similar, anyway. Would that make a difference?

I wouldn't be surprised if the purpose of an Akatosh priesthood would be to keep the other priesthoods in line by reminding them just who the king of the gods is. No other reason necessary; they just want to make sure that no other god usurps the Time Dragon's power. Perhaps I'm bringing too much Greek myth into it for that conclusion, though.