This seems like one of the biggest cultural differences between America and Australia (two otherwise quite similar countries) to me. Over here, if anyone is the most privileged, it's atheists/secularists for sure. If you're a Christian, Muslim, or anything else, you tend to keep it to yourself and try not to stand out too much. Probably only the Buddhists feel unthreatened, because Buddhism is seen as kind of cool, and tastefully non-committal.
I think that largely comes down to the different way our countries have evolved. Your country was formed by religious fanatics in an atmosphere of giddy idealism, while ours was formed by convicts and cold-hearted industrialists in an atmosphere of opportunistic pragmatism.
But in this one area, I think Australia offers a snapshot of the U.S's future. It's unlikely that the status quo will remain in the U.S. for long, and it's only a matter of time before you become more like us. All those moaners on the Christian Right who warn about 'Christian America' being under attack are, in a sense, right to do so. Their privilege is slowly slipping away from them, and they can feel it. Whether this is all a big liberal, pinko, satanic conspiracy, or whether it's just the result of people waking up to conservative Christianity's uncompelling message and clownish messengers, is of course a matter of interpretation.
I think that, like you say, part of the problem is that many American Christians can't see their own privileged status. But I think that the problem is also partly that many of them can. And they don't hold onto this privilege as someone who scored a lucky seat on a packed train and is reluctant to give it up. They hold onto it bitterly, the way someone holds onto a throne that they feel is about to be usurped.
Many of the greatest atrocities are those that are committed when the powerful believe that their power comes from a higher place, whether that be Heaven, Pure Blood, Destiny, or some other lofty and nebulous ideal. The strong belief that privilege is *rightfully theirs* is what opens the door for Nazis to massacre non-Aryans, Whites to enslave Blacks, Jews to occupy Arab lands, the U.S. military to bully whomever they please, men to treat women like property, and humans to enslave and torture animals in science labs and factory farms.
In each above case, both the severity of abuse, and the difficulty of ending it, stem from the fact that the perpetrators believe their superiority to be self-evident. (That's the great thing about things that are self-evident: you don't have to prove them!)
That's certainly the case with many Christians. Whether they see non-Christians as heathens, Satan's pawns, misguided fools or simply 'not-yet-Christians', the superiority of the Christian standpoint is clear, and to willfully cede that superiority would be to belittle, or even betray, the mission that Christ bestowed upon them.
And who can blame them? The Bible doesn't make a compelling case for pluralism and treating everyone's values and beliefs with equal respect. Though it may frequently ask for humility and compassion towards others, it also frequently trips over its own advice and talks about the special chosen of God, the absolute correctness of Christ's teachings, and about those who refuse to heed to God's word as being unworthy enemies of God, unforgivable, and/or needing punishment.
Eventually, the privelege of American Christians will wane, the same way it has in Australia and much of Europe. The tasteless hijinks and ugly, simplistic theology of conservative Christians will continue to alienate and embarass more and more people, until the privilege of a conservative Christian majority erodes into a small, diehard minority. I believe there's no doubt that this will happen, but I guess the question is: will the process be a relatively uneventful one of gentle erosion and mildening of opinions, or will there be vicious kicking and scratching along the way? It's America, so my money's on vicious kicking and scratching.