can we finally lay this particular linguistic lie to rest?
It is JUST NOT TRUE that "he" is a "gender-neutral" pronoun in English. It never has been true. English HAS no gender-neutral pronoun; it lost it centuries ago in the transition from being German.
It IS true that if you don't know the gender of the person you're talking about, you might say "he" -- but that's not because it's "gender-neutral". It's because the GENERIC human being is male, in people's heads. This happens at a very basic, unconscious level, and it's reflected in a lot of ways. This is one of them.
Let's give some example sentences to prove my point, and disprove this old canard. Note that the asterisk (*) is a convention in linguistics to mark an "ungrammatical" or "nonconforming" sentence -- that is, a sentence that will tend to make a native speaker of the language make this face upon reading or hearing it:
Each student should retrieve his books before leaving the classroom. Okay, so far, so good, right? That looks a little archaic, but doesn't violate the idea that "he/him/his" COULD be gender-neutral, right? But try the next one: *Every single one of them had a run in his pantyhose. This is pretty odd, huh? Still, we could be talking about drag queens or some sort of mixed group of dancers. But the last one is really the kicker: *I think everyone in this room is having his period right now. If you said this, a native English speaker would look at you like you have two heads. It FEELS WRONG. (I mean, you could have an entire room full of trans men. But even then it would FEEL wrong to most native English speakers.) This cognitive weirdness, the "huh?" you get when reading or hearing it, is precisely because "he" is NOT a "gender-neutral" pronoun. It is used for a "generic" human being, but that "generic" human being is always -- on a very basic level -- ASSUMED TO BE MALE by the culture and by people in it.