No.
"French" is adjective or a collective noun, but don't use it as a countable noun.
Trying to say "as a French" makes about as much sense as thinking "as a American" is correct.
As has already been said ... "a French (wo)man","a French person","a French citizen" is the correct way to go.
The reason you can say "an American" is because America starts with a vowel.
Same reason why you would not say "a British" but you could say "a Brit".
a French; an American; a Brit, or a British
sounds casual but correct to me
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.”)