If pronouns are such a problem for you and your daughter, how come you've referred to her as she/her throughout this post? Could it be that those are her pronouns?
Pronouns themselves aren’t the problem, Blueberry, as I’m sure you know. OP doubtless knows better than most that her DD is female and therefore it’s pretty obvious that she would use female pronouns for her.
Once again (yet again!) the problem is with normalising the use of pronouns that are used to designate people of the opposite sex to that which the person being referred to is; normalising the idea that someone who is clearly male can and should be seen as a woman if he so wishes; normalising the idea that everyone has a “gender” which is solely determined by how they feel about themselves (and apparently entirely based on regressive, restrictive stereotypes) - and that this “gender identity” overrides biological sex. And the overriding of women’s and girls’ boundaries, the male appropriation of things set aside for women as the more vulnerable and disadvantaged of the two sex classes that that entails.
Why is it so hard for you to break the habit of male oppression of females?
I can only conclude that you don’t truly see (biologically) female people as fully human, that you don’t see the suffering and oppression inflicted upon female people by male people across the millennia the whole world over as human suffering. It really doesn’t seem to matter to you. Perhaps we really are just service humans in your eyes, here to facilitate the lives of (biologically) male “first sex” people.
Because I can’t see how anyone who really took in and understood the wrongs that have been done to women (aka biologically female people) across the ages, and are still being done, by men (aka biologically male people), and who cared anything about social justice and wanted to see the world a better, kinder, safer place for humanity in general - I don’t see how anyone in that category could even begin to think of opposing any of the things that go some way towards redressing the balance for women in this misogynistic, patriarchal world, such as single sex spaces, services, and sports.
I don’t see how anyone in that category could not want to provide women (biologically female people) as a class with the maximum of safety, of opportunity, of anything that would mitigate against the great harms that have been done and continue to be done to women on the basis of our sex.
But breaking down women’s rights to those things seems to be your sole objective, by any means necessary, including promoting this “preferred pronoun” culture, especially among young people.
It does not reflect well on you, IMO.