Demonyms don’t use the same rules as countable nouns. Both “French” and “British” are acceptable demonyms, they’re just not particularly idiomatic in American English (which likes to overcorrect with “person” like you’ve noted).
(There’s no particularly consistency with this, it’s just what sounds “good” to American ears. We’re perfectly fine with “as a German” or “as a Lithuanian.”)
(There’s no particularly consistency with this, it’s just what sounds “good” to American ears. We’re perfectly fine with “as a German” or “as a Lithuanian.”)