A priest is a servant in the house of the god. It's a particular kind of work, implying nothing about spiritual purity or personal relationship with the deity.
The ancients did not have any sort of ministering to the ordinary people; such concepts, when implemented, are not strict reconstruction, but rather a borrowing from mainstream social expectations of what a religious worker does.
Darkhawk, an essay of yours comes to mind. Do you mind if I link to it?
I agree that priesthood is a specific calling, and does not necessarily make one closer to the gods than someone else. I think it is harder for us to define in our modern times, because nobody can do it full time. Unless we're independently wealthy, we have to work. None of us have a big temple to practice in. Most of us just have little shrines or alters in our homes. So what sets a priest or pristess aside?