you will not object to my pointing out that most U.S. citizens are indeed Americans, as are Argentinians, Brazilians, and even your ruthlessly polite neighbours to the North.
Of course. My main objection to US citizens calling themselves "Americans" is that they tend to feel they somehow own the designation. The term American is so broad, as you point out, as to be meaningless. It means anyone in N, S, and Central America.
I am also not a fan of words that are so broad that they are very difficult to understand without extensive context - unless that context is already well understood. On an international forum such as this, it is unclear at best and arrogant at worst for a US citizen to insist on calling themselves Americans. On a streetcorner in Iowa, it is much more specific - they mean "born in USA."
P.S. It is often fun to accuse the USA of being an "empire." I can't help but think this is tongue-in-cheek by critics. If the US had wanted an empire, we'd still own the Philippines and Cuba. We'd be "administering" places we occupied during WWII. Not to mention various and sundry banana republics. However, we looked at the British colonial model and decided we didn't want it. The US has been historically isolationist. Not even the harshest critics can deny that. So how do you square "isolationist" with "empire?" Answer, you can't.