'Singular "their" etc., was an accepted part of the English language before the 18th-century grammarians started making arbitrary judgements as to what is "good English" and "bad English", based on a kind of pseudo-"logic" deduced from the Latin language, that has nothing whatever to do with English... And even after the old-line grammarians put it under their ban, this anathematized singular "their" construction never stopped being used by English-speakers, both orally and by serious literary writers.'
The same reason it ever mattered how you refer to people, politeness and respect. If someone you consider "him" asks you to refer to them as "her" it's like someone asking you to call them by their full name "Rebecca" instead of "Becky" or "Jonathan" instead of "Jon". If you like and respect them, you do as they request because things which matter to them matter to you, and being polite to them is important to you. If you ignore what they ask, call them what you want, you communicate that you don't respect them and don't want to be polite, that you want to dominate and 'win' instead.
> "Pronouns can mean whatever you want them to mean"
Only one way. A specific person asking you to use a specific pronoun for themselves is wildly different from you unilaterally and universally saying that all women should feel included by the word "him" because "him" has no meaning anymore.
