“American” is the official English word when referring to people or things from the United States, but it heavily implies that it means either all of North America or all of North and South Americas. Most other languages have different words for American (country) and American (continents). If there were a campaign to replace the word “America” with something else when referring to the US, what would you think of it?
Another thing is that looking at the level of continents takes away from the nuances of situations. For example: CGP Grey talked about how most native peoples located in where the US is now actually prefer the term Indian compared to Native American, because Native American is a blanket term for natives on both continents, and includes too many very diverse and different groups to really be relevant to individual communities.
Same with Europeans, where you have significant differences between Western, Central, Eastern, and Nordic Europeans. Asians and Africans too, and to an even greater extent.