Too often in these sorts of discussions, the fundamental purpose of
language seems to get forgotten.  That purpose is to place thoughts in
the minds of others with as much precision and elegance (in both senses)
as the structure of the language allows (assuming an ideal listener or
reader); it is not to make a fashion statement or prove one's solidarity
with some social bloc or any other such thing.

In that vein, I believe that the immodestly named Walker's Law of Least
Cervical Displacement is a powerful guide: always use your words so that
the least number of listeners or readers are likely to jerk their necks
on hearing or reading them.  That, I think, will serve better than some
argument over what the received standard might be.

If I say "Each student was blowing his or her nose," not everyone will
find that perfectly felicitous, but it is unlikely to make any necks
actually jerk; but if I instead say "Each student was blowing their
nose," there is going to be some fraction of my potential audience or
readership that will, indeed, jerk necks.  The size of that fraction is
immaterial, unless one wants to claim it vanishingly microscopic, so long
as it exists; if one usage jerks necks and another doesn't, or one jerks
more necks than the other, the rule is clear on which to use.

All that is presuming what I stated as the assumption: that the goal is
to place thoughts.  If it is, any neck jerking, no matter how mild, or
how well the heads on those necks indeed understand you anyway, is a
speed bump on the road of smooth communication, and thus to be avoided.

--
Cordially,
Eric Walker