Interesting thread! I would add to this discussion the point that an outsider's experience of America to a certain extent depends on where you visit. Regional differences matter in some cases (for instance with how nice people are). Spending a week in Seattle versus a week in South Carolina will yield very different experiences on some of the points people discussed above. New York City is a wonderful place to visit, but as I have told other visitors in the past, If all you ever saw of America was NYC I don't think you would get a true sense for American culture and society in general, because I don't think NYC is illustrative of America as a whole. It's a very specific experience.
Now, having said that, if I had to suggest one place in America where you could get the broadest possible experience of America as a whole in one place, I would probably suggest Chicago: it is a large cosmopolitan city, so it is representative of the large urban American experience in general, it has strong working class roots and still has a working class ethos that permeates the city (which some places in America don't have yet some do), it simultaneously has lots of diversity (for example the Albany Park area has something like 84 different nations represented in its citizens) but at the same time there are a lot of "white" people too (the generic mid-western type which in my mind is the closest thing to the default type of person most common in our country), it has a lot of what is typical of suburban sprawl (new suburbs with the generic white picket fence feeling to older suburbs with the brick and mortar type of home), the economy has a very broad spectrum of all things typical of America, etc. I also feel that Chicago displays both the best and worst of Americans on a singular, one-on-one level too (people are nice and polite in the more common ways but they are also used to a very fast paced and aggressive city lifestyle, so you also get to experience the many rude things Americans typically do in such places).
On the other hand, I can think of many things that occur in Chicago that don't occur in other places, so I guess there really is no substitute for visiting lots of different places here if one wants to have the comprehensive and complete American experience.