You're missing a key detail about how the world of Ice and Fire works. Namely that:

Old Gods, Drowned God

Those are different religions from the Faith and they only survive in specific areas of the Seven Kingdoms.

The Faith is set up as the Westerosi equivalent of the Catholic Church, with their own cardinals and the conclave (the most devout), and is supposed to have this tremendous importance on Westerosi politics. But when examined up close, the Faith has none of the actual traits that the real-world Church had to exert its enormous influence.

The Faith seems to take its power from the will of the omnipotent will of The Author, rather than in-universe mechanisms.