There are multiple usages, specifically just citizens of the USofA is just one use.
American, a. and n.
(əˈmɛrɪkən)
A adj.
1.a Belonging to the continent of America. Also, of or pertaining to its inhabitants.
1.b American language (usu. with the), (i) a language of American Indians; (ii) American English (see sense 3). Also American tongue.
2.a Belonging to the British colonies in North America (obs.).
2.b Belonging to the United States.
2.c U.S. spec. (See quot. a 1861.)
1837 Diplom. Corr. Texas (1908) I. 187 A large number of fine American horses‥which there is no doubt had been stolen from citizens of Texas. 1846 E. Bryant What I saw in Calif. (1849) iv. 37 Such [Indians] as rode ponies were desirous of swapping them for the American horses of the emigrants. a 1861 Winthrop John Brent (1862) ii. 14 He was an American horse,—so they distinguish in California one brought from the old States. 1878 J. H. Beadle Western Wilds, xvi. 253, I rode a good-sized American horse.
3.a Special Combinations. American bar; American blight; American cheese; American cloth; American dream; American English; American football; ...oadfoot et al. Billiards i. 41 In 1876 D. Richards‥ran second to Cook in an *American tournament. 1976 Cumberland News 3 Dec. 19/1 On Thursday, December 16‥a Christmas American tournament will take place.
3.b In the names of various trees and plants native to North America, as American arbor vitæ, Thuja occidentalis; American ash, Fraxinus americana; American aspen (tree), Populus tremuloides; American Beauty (rose), a variety of cultivated rose; American beech (tree), Fagus grandifolia; American elm (tree), = white elm; American plane (tree), the buttonwood or Virginian Plane (see plane n.1 1).
B n.
B.1 An American Indian.
B.2 A native of America of European descent; esp. a citizen of the United States. Now simply, a native or inhabitant of North or South America (often with qualifying word, as Latin American, North American); a citizen of the United States.
B.3 A ship belonging to America.
B.4 pl. Short for American stocks or shares.
B.5 American English; the form of English spoken in the United States.
Yes, that is one use of the demonym “American” in a modern, geopolitical context. There are other contexts and other uses when it has different meanings.
Asserting it has that meaning when used in context of a time tens of thousands of years ago is nonsensical.
