Un Océano Inmenso CaZIH sin Olas

While you are correct, I suspect that a person from Zihuatanejo or a person from Toronto, should they identify themselves as “American” in Europe will be treated less kindly than they would be if the identified themselves as Mexican or Canadian respectively.

The point being, we don't use it to refer to a nationality.

During the Paris Olympics, for example, a Mexican watching other athletes would likely feel more drawn to root other Americans, even if they weren't all Spanish speakers. It's how we commonly refer to our collective selves.

Even a U.S. passport under Nationality does not say "American".