We believe in a common heritage and we think that because our ancestors were pioneers in so many aspects, from mathematics to theater and from philosophy to commerce that the whole world owes us some (actually a lot) respect.
The mindset you're referring to is true but applies to the state. We all thing that the government in particular and the state in general is crap but as far as we're concerned as individuals or as a nation we walk in the footsteps of our glorious ancestors.
And it's exactly this mindset which have led to our recent problems, which are not only financial as many believe. If you keep thinking that you are someone special there's no reason to change anything.
That doesn't follow. The fact that some people say "we were civilized before you" doesn't mean they think it's the best country in the world. Not to mention that nobody even really says that, unless they're looking for a cheap insult. It's the equivalent to the American "we'll bomb the shit out of you".
That sounds absurd to most Americans too. I've only heard that kind of rhetoric spoken by politicians, and it's definitely not a view most of us espouse.
There are a large number of Americans who believe the U.S. has the most powerful military in the world, and there are a sizeable number of Americans who believe that the U.S. provides the most opportunity for poor people to become rich through hard work (the "American dream"), but there are at least as many Americans who disagree with both, and I've never heard anyone in real life state that the U.S. is categorically "the best country in the world." Interestingly, I think it's safe to say that almost all Americans hold one view in common: no one likes the central government. That might be true in most democratic countries--I wouldn't know. But I don't think I've ever in my life heard an American state that they were satisfied with the government.
As a person that lives in the Bible Belt, I can tell you that a significant number of folks that do not think America is the best. Many do believe it was at one time, but some of the strongest critics are in the Bible Belt now. The Tea Party movment, while arguably misguided, is a great example of that.
Seeing Americans in movies go "best country in the world!" sounds absurd to us, and "you can't do this to me, I'm an American" sounds ludicrous.
The fact that everyone considers their country to be the best really seems to me like an American generalization. I've lived in the UK and I've had friends from a multitude of countries, and not one of them considers their country to be the best, unlike my American friends.