Afghan schoolgirls are weeping as they finish sixth grade, knowing their education is over. Under Taliban rule, they are unlikely to ever step foot in a classroom again.
It isn't about religion with any of these types. Rather, it's about religion being a tool they can use so that those who do think it's all about religion allow them to have authority.
Religion is the second worst plague visited on humanity. As you say, the worst is the lust for power over others. Religion, especially the Abrahamic religions, is second because it is uniquely positioned to facilitate the ambitions of those who lust for power. Pure violence can subjugate a population for a while, but real, long-lasting, self-regulating, trans-generational oppression has usually required religion.
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.