Monica, you have IMHO correctly described the psychological origin of "priesthood" as it arose in many pre-Christian cultures, especially if you add offering sacrifice on behalf of the un-ordained to the mix. But it is my impression the for most of the first millennium, our understanding of ordained ministers that eventually settled down to bishops, presbyters, and deacons was precisely different in nature. 

The Latin and Greek terms for priest (sacrrdos and hiereus) were avoided. It was I think the transformation of the theology of Orders and a particular understanding of the Eucharist as the sacrifice of Christ on the cross that led to an understanding is our sacramental priesthood as a m"priests" in the primordial sense you delineate so well.

One minor quibble, outside of Christianity, the notion of fallen nature is not required to set up the idea of holy priests/priestesses as mediator between the gods and their supplicant people.