The neutral pronoun in English is 'it,' and using it to refer to a human being is always wrong.
The use of singular they is in a few contexts correct, but in many more contexts incorrect. The use of 'he-or-she' is inelegant and incorrect, and the use of generic she is less incorrect than silly.
Seriously, this is remarkably silly. 'Girl' in German is a neutral word, but no-one would say that German girls think of themselves as genderless. 'He' in English is normally the correct word to use generically; no-one sane would say that English-speakers think everyone is male.
(for that matter, the English word 'man' actually is genderless: the English word for a male human being is 'were,' as in werewolf — it's cognate to Latin 'vir')