It's not completely wrong, it will be understood, but it is ungrammatical and a clear marker that the speaker is not native, similar to getting adjectives in the 'wrong' order ('a big tasty sandwich' sounds more natural to a native speaker than 'a tasty big sandwich', even though the latter makes sense and will be understood).

Demonyms for historical neighbours of England have irregular forms when speaking of a particular person from there. Scotland has 'Scot' and 'Scotsman'; Wales has 'Welshman'; Spain has 'Spaniard'. Other countries indeed need a second word, such as 'person' or 'citizen' ('a Chinese' sounds offensive to me; I would say 'a Chinese person' in all cases). The only country I can think of where using a bare demonym is grammatical when speaking of a single person from there is Germany with 'a German' - probably because it has the suffix -man.

Edit: A sibling comment pointed out that 'an American' is grammatical, and thinking about it, I think the suffix -an is what makes bare demonyms grammatical - you can say 'an Angolan', 'a Laotian', 'a Peruvian', 'a Moroccan', etc, but wouldn't say 'a Thai', 'a Swedish', 'a Sudanese', etc.

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".

Technically yes the demonym is "French", but "I'm a French" just doesn't work in English. The word 'French' is almost exclusively used in English as an adjective or the name of the language. It is never used as a noun for anything else. So in context, it reads as an adjective without a paired noun.

In English, you have to disambiguate be adding a noun: French person, French citizen, or Frenchman if you're old and inconsiderate.

Similarly, we don't call people "a Chinese". That construction is considered derogatory, if not outright racist. Demonyms typically cannot be used as nouns alone without a suffix. "A Brazilian" or "a Spaniard" are acceptable.

As usual for English, the rules are vague and inconsistent.