I largely agree with the things you've said above, but I'm not the one correcting papers or writing style guides. Anyway, most everyone will get there eventually, just not yet.
Woods wrote:So what's the alternative?
Well, "he or she", "he/she". I've occasionally come across the concepts of using one's own gender pronoun or alternating between "he" and "she". You can also restate the noun, though that risks sounding repetitive. The strategy that most people seem to agree on is rephrasing the sentence if possible to avoid having to make the choice at all. That could mean making the referent plural, for example.
Woods wrote:where is the limit between formal and informal contexts, i.e. from where on should I go he or she instead of they?
There's not a single, neat answer to this question. As I described, the consensus on singular they is in flux.
The Associated Press, which publishes a style guide used by a lot of U.S. news outlets, changed its guidance a few years ago to allow singular they "when alternative wording is overly awkward or clumsy", but it strongly encourages rewording (
link). The MLA style guide, commonly used in academic writing, does not use singular they, preferring rephrasing, or alternatively "he or she" (
link). Both allow "they" when it is a person's gender pronoun (though AP specifically disallows neologistic pronouns like
xe, while MLA implies they are acceptable).
And that's just two specialized authorities on usage. I don't know what contexts you use English in.
N'hésite pas à corriger mes erreurs.