I don't disagree with you but I think it's important to draw a distinction between the motives for that influence, which I believe falls primarily between three possibilities: someone seeking to save souls (or true believers with no personal motive other than maybe getting points for their own soul), someone seeking to manipulate others into being good to each other, and someone just seeking personal power and who will adjust the "good" their religion supports based on their situation.
They can all blur into each other, so reality is a bit more complicated than that, but what I wanted to express was that most (if not all) religious leaders present as the first, but some not insignificant portion of them (if not all) actually fall more into the 2nd or 3rd categories. They don't believe in the same god that their followers believe in.
I think most assume that they are true believers.