The difference with pronouns/gender is that they depend on and proclaim external facts. ("She" is used for women and by going by she, I am a woman.) And by asking someone else to say that I am a woman, then really, aren't I forcing someone to change their definition of "woman"/"man"? Why do I have the right to force someone to do that?
Out of curiosity, if someone asked you to use plural pronouns for them, would you? (This is hard to ask in English, because "your" is used for both singular and plural subjects, and "they" has also commonly been used for both singular and plural as well, but imagine) What about pronouns in other languages? Do you really think that in communication, the other speaker shouldn't have any say in any of this?
I don't appreciate comments that suggest that someone you're discussing is an "asshole" for doing something you disagree with and don't think that contributes to the conversation.
If your gender is like your name, which you can choose yourself, then it can't be bound to anything you can't change, like your chromosomes or the hormone levels you had during puberty. In other words, it can't map to biological sex. Which, okay, that's kind of the idea, isn't it? Someone wants to have sex male but gender female.
Except that then nobody really cares about gender at all. When you want to have things like women's sports or prospective dating interactions in which people may want to have biological children or anything else where sex differences actually matter, what people want you to declare is not your gender but your sex.
This then becomes the motte and bailey where gender is supposed to be different than sex but somebody who has reason to actually care about sex is then told that they have to accept gender instead or they're a bigot, as if they aren't different. And that is the point of political contention.