Using "she" when the gender is likely to be female, and "he" when unknowable might have been ok when 99% of all nurses and midwives were female, and 99% of all doctors and engineers were male, but that is no longer the case. It doesn't matter whether you approve or disapprove of that social change, the fact is that the old times are not coming back. Here and now using gender-specific pronouns with reference to gender-neutral titles is becoming increasingly bothersome, confusing and embarrassing. Using the gender-neutral plural pronoun very neatly gets round the difficulty, even if it is strictly grammatically incorrect. It certainly seems to be the way modern English is going. I'd rather that than saying "he or she" and "him or her" all the time, that's for sure.
Ah, OK.
So we agree that “he or she” is mostly unnecessary. although, ironically, right before I read you post this morning, I made another and used “he or she”, in the following context: the addressee was clearly a woman, judged by the screen name, but in my post I was making a brioader reference, which, however, was pointed at her. Hence I couldn’t use “he” alone.
I think structuring the phrase for plural is fine when it comes naturally, but people should not strain to do so. “If they participate in the raffle they should be proud” is fine. “They who win this raffle will have a nice prize” is strained and misleading, the winner is usually one person. “The winner of this raffle will take a grass mower home with them” is definitely not fine, it violates the English grammar.