I don't trust straight democracy either, as it can turn into mob rule. And our founding fathers felt the same way, which is why we have a republic. But the neocons are wrong if they think a state-sponsored religion does anything but make people cynical about both the religion and the state (there is a whole strain of Muslim humor sort of like blonde jokes that present Imams as corrupt freeloaders.)
This country used to have, for want of a better word, a common "secular" culture or "civic religion" as I think Kristol put it, that agreed on "Amercian" values (hard work, fair play, tolerance in the un-PC sense, neighbors helping neighbors, etc.) In that sense, Americans did have shared values, compatible with an amazingly wide range of religions and cultures, that bound us together. If an immigrant came here and agreed to abide by these values, he "became an American" -- a cool way to put it, we have always thought being an American was like joining a club, not an accident of birth. Unfortunately, these "American values" have been systematically undermined by the PC crowd, knocking hard work if somebody on welfare is offended, fair play if somebody wants affirmative action, and on and on. I think this is the biggest red/blue divide: what does it mean to be an American, and is it important? But that's a little off topic, just note that Americans specifically believe in freedom of conscience and religion, and for the majority of the religions we've had in this country it's worked for us.
But because it this "civic religion" has been so successful, I think Americans have fallen into a trap that all religions are compatible with being an American. And that's a fallacy, somebody wanting to restore Aztec human sacrifice would come to the attention of the local district attorney pretty quick (except maybe in San Francisco!)
So is being a Muslim compatible with being an American? Me personally, I don't think so. The simple historical fact is that Islam has always spread through conquest, which Christianity by and large has always spread through conversion. I've read the Koran and some of the Sura -- it doesn't seem like there's anything in them to hang seperation of church and state on. Can the Muslims have a "reformation" and define which texts matter and in which ways? I don't know -- I don't see an "opposition" Muslim movement speaking out aggressively against the Wahhabbists and other radicals. There seem to be either apologists for the terrorists or Muslim groups in the west that maybe put out a press release geared towards non-Muslim consumption. If there is going to be a "protestant" Muslim movement, it has to be start inside Islam, be "on fire" that they are right and speak to fellow believers. I'd be thrilled to see it, but I'm not betting the rent money.