I think you've got it wrong: priests and bishops raise levels of their Christian faith. Provinces have differing levels of faith in the various religions, and these pieces are useful when you need to align newly-conquered regions to that of your leader; indirectly increasing happiness.

IIRC it's inquisitors and cardinals that raise zeal, at least until they start burning people. They can be used to get rid of generals with low piety. Zeal is what makes people join a crusade when it passes through. It might also affect the nature of an uprising, should a province have cause to rebel, ie. you get Christians revolting rather than just peasants.

Thus, priests/bishops prepare the ground before an invasion or help to integrate new territory. Inquisitors/cardinals can be used offensively to cause problems for the AI.