There is no "free world". It's a myth. Some other posters noted that too. There are only relative degrees of more or less freedom here and there, a mixed picture everywhere, which is to say...some people have a bit more freedom in this or that respect than others do, but NOBODY is completely free.

The concept "leader of the Free World" is rooted deeply in past American mythology (and believed by many Americans....but not believed by a good many Americans now). You'll hear it repeated again and again by politicians and commentators, because it's a cliche, a standard piece of politicial rhetoric.

As others have noted, America's present hubris arose following a lengthy period of really extraordinary hubris on the part of the enormously successful British Empire. The British, in their own inimitable fashion, were just as arrogant in their assumptions of innate societal superiority over others for several centuries, but they kind of ran out of money after the two World Wars, and the USA quickly supplanted them as the foremost superpower. So it goes. Winston Churchill must have found it profoundly disturbing to see it all slipping away! ;-) Those last few mighty dreadnoughts of the Royal Navy proceeding mournfully to the breakers' yards in the 1950's...while the US Navy patrolled the oceans of the world with its new fleets of modern aircraft carriers.

Well, all things pass, don't they?

The French have also enjoyed periods of great national hubris, most notably under Napoleon when they rather briefly dominated almost all of continental Europe.

The Russians have enjoyed periods of great hubris. So have the Germans and Italians and Japanese, but only very briefly.

So it goes. They all had grand pretensions when their tide was running highest.

The use of the term 'the Leader of the Free World' by US media commentators is ridiculous....but don't expect them to stop using it. ;-) They're reading off a very old script that has been in use for so long now that it's become like wallpaper. They say it without giving it any real thought, I would assume...or else they're living in a perpetual state of fantasy.