I think a more accurate analogy is between "Eurasian" and "American." How often do you use the term Eurasian?

In practice, most references to the people on the American continents are to North Americans, Central Americans, or South Americans. Referring to the collective as 'Americans' is rare and virtually meaningless, given the total lack of shared culture, language, or ethnicity between say, Canadians and Brazilians, comparable to comparing Britons and Chinese.

And in that rare case, context should make clear whether you mean 'people of the Americas' or 'inhabitant of the United States of America.'

It really isn't an issue and I've always been baffled as to why anyone should have problems with the demonym American as it is popularly, and correctly, used.