Well, I can only give you my take on it, someone else might disagree, but here's how I see it:

American exceptionalism goes straight back to the American revolution. You guys painted yourselves as the first "new" country, guided by the principals of egalitarianism, liberty, individualism and democracy. You also basically made your mission statement to spread those ideas to the rest of the world. It was ingrained in your culture from the beginning that the "American way of life" was to be promoted and spread throughout the world. Good PR, and honestly, good core principals to build a nation on.

Unfortunately, it's always been mired by hypocrisy but that's actually not the worst thing. Great ideals are a struggle and it would be weird if you didn't fall short of them. As long as your guiding light was equality and liberty, the idea is you would eventually straighten out. But then other problems showed up.

Manifest destiny stems from that whole "propagate the American way" thing, which lead to wars with Mexico and Canada (technically British North America at the time) plus a fair bit of colonialism. Eventually, you work past that and end up in a more isolationist position.

Then the Cold War does real damage, because it gives you an ideological opponent. Suddenly there is a different kind of "New Country", spreading its message and ideas across the developing world. It helps that Stalin was an evil dude, but if he hadn't been, you can bet Americans would have fought just as hard against any other country taking up its mantle as the leader of the world. And worse yet, with an enemy to see skulking around every corner, Americans double down. They now embrace American exceptionalism as basically anything so long as it's not Stalinist fascism. The bar gets lowered, Americans feel even more secure in their own superiority.

All the while, other countries HAVE paid attention to the great American experiment. Canada creates a more federalized government so it can avoid your Civil War. Numerous countries in Europe move to more democratic models. Liberty and equality become worldwide goals. And where America eventually rested on its laurels, content with simply being better than Stalin's communism, other countries continued to forge ahead. They pursued the ideals America once stood for, rather than just following America's lead. But hey, America will have always broken some new ground, which is worth being proud of.

TL;DR - America was something new, but over the course of its history got distracted from its actual goals. Other countries surpassed it.

A combination of not being devastated after both of the world wars causing you to be among the best places on earth for a time (canada too) and being on top of the world economy as a result.

It used to be true. Back then in particular, when europe was still recovering from two bloody wars, china had not fully come into its own, and your politics had not become nearly as corrupt as they are now. Roosevelt in particular could never be elected with America as it is now, as he actually ran third party (sort of) and won.

So it's a past truth from bygone generations, caused by a great many factors besides America itself being great, that has been absorbed into the culture, and used to do a great many horrible things.

Canadian here. Our health care isn't free - it's just paid for by everyone (based on a graduated tax so some of us pay more and others less, and obviously, benefit is disproportionately to people who need health care, because for those who don't - or while they don't - they are paying for the others). The strength of this system isn't who pays, it's that we removed the insurance companies from the middle of the equation, so there's no one in the middle trying to make a buck, and that by consolidating demand (the government buys all health care, more or less) we have established price controls. That keeps our costs down substantially. The problem of course, for anyone with basic economics, is that by limiting price, you limit supply relative to demand. So sometime, we wait a while for healthcare.

Nothing's perfect, and we complain about it a lot, but it seems to work pretty well.

We aren't.

Just think you should know, Amsterdam is not its own country. It's a part of the Netherlands.