I agree that a 1 policy faster great prophet would be a good solution.
For faith, I'm not willing to remove faith pantheon that seems way too harsh. I don't want to make religion a monopoly of Piety, I want Piety to give the most out of Religion. It already has some strong bonus for spreading (better faith, the mosque and a discount). The little bonus to the capital in the opener would be there only to ensure an early pantheon which seems like a good idea. Limiting something like Desert Folklore is however a possibility.
An option I have been toying with would be to offer Piety a unique building available through faith purchase that would increase Happiness and or culture. I think the tree need some form of happiness and giving it through faith seems a good idea.
It's actually 2 policies faster since you don't need Theocracy.
But it's not about Piety having a monopoly (Though they should get first crack) it's about ensuring that you need to make an investment to get a religion. With the right terrain you can reliably get a religion by building a single shrine, and I've had games where I didn't even need that. Hit two Religious CS next to each other with my warrior and picked DF. So literally 0 investment and I get the "best" religion. So the problem is that I get all these bonuses from religion at near 0 cost. If instead I had to build shrines/temples to get a religion that single shrine will get me a pantheon and that's about it. Which is how it should be, a small investment gives a small bonus, a bigger investment gets a bigger reward.