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

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.

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.