Weird fun fact (as an Argentinian who went to school in England for a few years): in English-speaking countries, America is not a continent in the same way as in Spanish. In English they have two continents: South America and North America.
So the word "American" in English does not mean the same as "Americano" in Spanish.
There's really no natural word in English to refer to someone from "El continente Americano", because no such continent exists in English. That's why they use the word "American" to refer to someone from USA exclusively.
That sounded fascinating as a rather large difference in world view stemming only from using different languages.
It turns out that there are various models for the number of continents, and that is (phew) known in Spanish, too. See the Wikipedia page [1] (link to Spanish version) for instance. This is for European Spanish though, but I couldn't find a version of the page in es-AR.
I think "the Americas" means the continent(s), and America (to some extent) can mean either but it would feel more like something used as a gotcha at a pub quiz.
You're definitely right about there not being a word for someone from that continent though.
"American" to refer to USA exclusively does make sense either way because USA shares the continent with at least two other countries no matter how you slice it.
Frankly, the model with the single America continent doesn’t make any sense, because south and north Americas are so different in both geographical and cultural/historical sense.