It’s not about size. It’s the fact that the United States of America has the word “America” in it. And I don’t refer to the US as “America” (unless I’m being cheeky, though in those cases, I spell it 'Murica), but I do refer to people from the US as “American”.
And I know this is all kinda pedantic. I just think it’s fun to talk about words. I get the feeling you read some snark into my pervious comment, but that really wasn’t my goal.