the doctrine choices would have a combination of 'state religion effects' and 'non-state religion' effects I guess, so depending on the choices they would have different effect in opponent cities, or in your own if you don't use that one as state religion. The effects are tied to the religion, not the civ, so the effects of that religion are both freely visible to all civs, and is global, so there will be situations where players will be more or less concerned about different religions spreading in their lands or even welcoming of them. They may even decide that piggybacking and adopting another player's religion over one they founded has good benefits and strategic value in the short term. Some effects may only apply to the founding civ, but in most cases any civ where it spreads could take advantage of it if they wished.
Let's say Caesar and Harun are neighbors; Caesar has founded Christianity and Harun Islam. Both have their respective homegrown religions as state religion. Caesar's doctrine choices give Christianity +20% to production, and Harun's choices give Islam +20% to culture.
Now consider Ravenna, a Roman city with a population of 10 that's close to the border with Arabia. Harun has done considerable missionary work, and the result is that five of Ravenna's population is Muslim and five Christian. What does this mean for the city's output? Does it get +20% production, because Christianity, Caesar's state religion, is present in the city? Does it get +10% production and +10% culture, because its population is split between Christianity and Islam? Does it get +10% production but no culture, because its population is split and Islam is not Caesar's state religion?
Now imagine that Bismarck is also in the neighborhood. He founded Judaism, but chose to share Christianity with Caesar rather than face his wrath. So nobody is "responsible" for Judaism; nobody is selecting doctrines for it. What are the effects of Jewish citizens in a city? Are they simple liabilities, since the religion bestows no benefits?
Now add Elizabeth. She decides to convert to Judaism, even though she didn't found it. Does she get to select doctrines for the religion? What happens when Bismarck decides to convert to Judaism after all? Does control revert to him, or stay with Elizabeth?
Yeah, this is a good point and I agree and want to try and avoid that. I guess the balance would be to provide opportunities to change them, but make these a lot more confined than changing civics. Either by wonder / advisor effects or even a 'per-age change' or something? while still maintaining a more persistent model would provide a few moments for a player to adjust their strategy.
To be honest, my first thought when reading your proposal was that leading reformations sounded like the perfect job for Great Prophets.
But I think I have a better idea: gradualism. Civics let you jump from any option to any option with little difference except the duration of the anarchy. But doctrinal change is more often evolutionary than revolutionary; if you're Very Intolerant, for instance, you can only move to the "adjacent" doctrine in the Tolerance column, Sort-Of Intolerant. (Placeholder names, obviously.) The doctrines themselves would form more of a continuum than civics; while Mercantilism and Free Market have completely different effects, Very Intolerant and Sort-Of Intolerant would be similar, but different in strength.
And if you really want to make religion play different than civics, you could give
every leader who belongs to a certain religion the power to evolve that religion's doctrines. So Bismarck and Elizabeth could engage in a struggle of ideas over the direction they want Judaism to take, Bismarck moving it "up" some column and Elizabeth moving it "down". Maybe allow leaders with larger populations of the appropriate religion to make changes more frequently.