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.”)
> Both “French” and “British” are acceptable demonyms
No they are not.
The Oxford English Dictionary, for example makes it quite clear re. 'French':
I draw your attention to the first three words ... "with plural agreement".It is explicitly telling you that "French" is a collective plural noun and hence cannot be used as a singular countable noun.
I think we’re past OED being a normative arbiter of what does or doesn’t pass for acceptable English usage.