I must say, I find this idea of abstracting religious functions an intriguing solution to some of my problems regarding religion.
My main problem with using RL religions is, from what information we have so far and without having played it, it sounds like they've created an espionage system, slapped some religious names on it, and placed it totally in control of the player with nothing to differentiate one religion from another.
I think this could potentially solve the problem of the player completely controlling the religion without creating the opposite problem of taking the player out of the driver seat. By creating a "religion" screen, similar to the espionage screen, you could create the illusion that the player isn't so much controlling the religion as much as interacting with that religions head of state. Most of the interesting ways that religion interacted with politics was as a powerful extranational force. Sometimes their aims coincide with the rulers, but not always.
Hear me out on this.
You can contact the heads of all the religions that you currently have in your empire. You can induce them to conduct various religious missions, such as sending missionaries to a certain city(leading to more conversion - financed by gold), causing their believers in a certain city to agitate (increasing maintenance costs - financed by favor), etc. Everything would be paid for by favor and/or gold. You build up favor with a religion by building temples/shrines/cathedrals/etc. Some things could only be paid for with favor, some with gold, some favor supplemented by gold. The price would depend on how that religion feels towards you (depending on its traits, using Aussie's system).
Granted, some obstacles remain to make it seemlessly mesh with Aussie's system of play-defined traits or within a generic religion system, but I think this has a lot of merit.