I disagree with the premise here that "America" describes the continents of North and South America. That is in and of itself the issue here, so I don't agree that the USA is a subset of America because I would refer to the continents as "the Americas" or similar.
This shouldn't be mixed up with the other common case which is often discussed, 'American' as the demonym for people and things from the USA. That isn't quite as bad (though it still annoys other people from elsewhere in the Americas) because it's using the name of the superset - Americans, from the USA, are also Americans, from the continent. Whereas, when "America" (the USA) does something, it doesn't necessarily mean all of America does it. The USA is a subset of America, not a superset or the only entry.
Language isn't always logical, but if you won't draw the line at something so clearly backwards as using the whole continent's name for the actions of one country, you can only expect it will start an argument every time.