Counterpoint: there is no continent named “America.” “North American,” “South American,” and even “Central American,” or “Latin American,” for added specificity, are completely sufficient demonyms for the denizens of the continents (and subreigon) writ large.
I agree with all you’ve said, and I tend to add both systems when expressing a meaningful measurement. My statement is pointed more towards situations where someone hasn’t done so and it throws some poor soul into a meltdown.