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.
> and a clear marker that the speaker is not native, similar to getting adjectives in the 'wrong' order
I would think that if you say you are French, then everyone know you aren't native anyway. Maybe it's actually a good way, it can distinguish between true natives and false natives
> but it is ungrammatical and a clear marker that the speaker is not native
You mean a native speaker might be ungrammatical when using their non-native language? That makes sense to me.
> Spain has 'Spaniard'.
Even so, you'll hear a ton of native Spanish people saying "As a Spanish person" or "As person from Spain" instead of simply "As a Spaniard", I'm not sure this is very surprising. If anything, that mistake makes it more likely they're a native than not, in the case of Spain, as the level of English outside of metropolitan areas is lacking at best, compared to other European countries.
I'm using the words 'grammatical' and 'ungrammatical' in a linguistic sense; human languages are subtle and fluid, and one doesn't have to be far along the sliding scale between 'doesn't speak a word' and 'well-educated native speaker' to be understood. We speak of 'broken' English when somebody is able to be understood but hasn't fully grasped the language yet; using demonyms incorrectly is a subtler flavor of the same thing. For example 'no come here' -> 'no entering' -> 'no entry'
> but wouldn't say 'a Thai', 'a Swedish', 'a Sudanese'
You also don't say 'a Japanese' but that is an extremely common error with Japanese English speakers when they are first learning.
I am looking for a citation, but I seem to recall the casual rule of thumb is something to do with the ending of the nationality (so '-ish', '-ese','-ch' etc. you can't put 'a' in front). I think the more formal explanation likely centers around rules relating to indefinite articles.
There are some suffixed with "-i" which sound natural to my (American) ears too: "an Israeli", "a Somali", "a Pakistani", "an Omani", etc.
As a Welshman, I’d say North/South Walian are more common among the populace!
When speaking English, the French side of my family refers to themselves like that often, however, they're from Bretagne, so exactly how French they are is up for debate.