You make other good points about the problems with Haley's stance, but...what's this doing there?

If trans women (and men) were considered a "third gender" (or maybe 3rd and 4th), basically all of Haley's points would go away. It seems, logically at least, that she would totally be okay with this. She's entirely focused on gatekeeping who is allowed into the "girl" category, not on rejecting new categories, at least from what you've said and what I've heard.