However, I don't get why someone would ask themselves "what's so great about me that I think I should be able to communicate directly with God?" and come up with something they think justifies doing it.
I could concede that in previous times, considerations of social prestige may have been a factor for many a prospective priest. (Albeit at the steep price of a celibacy vow if you're Latin Rite, as well as the daily obligation to pray the Liturgy of the Hours). These days however, since there is little public prestige in the Catholic priesthood, I don't think arguments of self-interested motivation as being a major driver of vocations flies.
More to the point. It is absurdly reductionistic (and circular) to think that the only reason anyone would pursue the priesthood is narcissism. As hard as it may be for the enlightened atheist to grasp, some of us still have the audacity to believe in God. And some, are even willing to dedicate their lives in a very direct way for that belief.
Also as an aside, none of the apostolic traditions (Catholicism, Eastern Orthodoxy, Oriental Orthodoxy, etc) make the claim that the priesthood have direct communication with God. (You don't need to be a priest to pray). The priesthood exists primarily to teach the faith (which requires literacy and training), administer the sacraments and offer the sacrifice of the Eucharist.