You might consider giving a free advocate when a religious tech is researched (and not just to the first civ to do so). This would of necessity include an advocate at the start of the game. That way you don't have situations like the Klingons not being able to follow warrior code until they figure out how under the tutelage of the Founders, for instance.

Additionally, it makes sense to me to have other types of GP's able to build some of the shrines, rather than just prophets:
Merchants for Capitolism
Generals for Warrior Code
Scientists for Total Logic
Spies for Isolationism
Engineers for Expansionism
Artists for The Prophets or Pah-Wraith, maybe

Fall from Heaven uses both these mechanics, to give credit where due.

I can do that. I'm not sure about moving all the shrines to other GPs... some might fit better with the prophet than elsewhere.

Warrior's Code and Total Logic are tricky. Both are inspired by the more spiritual forms used by the Klingons and Vulcans, but their implementation is more general.