There was a surge in U.S. patriotism following the Civil War. Prior to that, it appears that people were more patriotic towards their individual states than to America as a whole. But after the Civil War, patriotism and loyalty to the entire nation was strongly pushed. I don't know if very many people genuinely believed that America was the "best" at that point, as we were still playing second fiddle to the (mainly) Anglo-French hegemony over the world. It wasn't until WW1 that we started to overtake them as a world power. WW2 seems to be another turning point, at least as far Americans' perceptions of being the "best."
Some of it, I believe, was probably spread on a bit too thick, perhaps to placate the egos of Americans whose worldview tended to be more provincial, insular, and isolationist. If Americans are led to believe that America is the "greatest" and absolutely "indispensable" to safeguarding the "free world," then they will be far more pliable and supportive of U.S. global militarism and interventionism. Because, "If not us, then who?"
If we were just an ordinary, run-of-the-mill country, then attitudes would probably change. Like, say if we were Paraguay. Do you think the people of Paraguay sit around fretting about what might happen if China invades Taiwan? I doubt it. But Americans feel compelled to take on the worries of the world on their shoulders because they believe we are the "greatest," and therefore, that's a great responsibility we must uphold.
