No one is saying they don't have rights. What they don't have is cross sex rights.
Because really, this cross sex identity, this living "as a.." thing, it isn't an actual thing in its own right. It can literally only exist in a framework of sexist gender roles, because those are the only things of "womanhood" that are ever accessible to men (and of course, this is also why these things are not in fact "womanhood" at all).
You are fixated on the solution here being to take something from women. You have been lead to believe the solution must be a "balance of rights" between women and cross sex identifying men akin to the balance of rights between a woman and the unborn children dependant on her body for life.
This is a false equivalence. The child has no option other than the woman's body. Cross sex identifying people however could have many options for dignity, safety and hell, just for support and fun that do not involve encroaching on the provisions of the opposite sex. They could already have had this had the pressure groups that are supposed to represent them not been so ideologically wedded to the single solution of cross sex colonisation, an obsession which has ultimately taken them to this.
So yes, absolutely the law should find ways to protect them, and the people who face these challenges should be an active part of defining that solution. The only constraint this EO introduces is that solution cannot involve cross sex appropriation of single sex provision.
It does not preclude additional mixed sex provisions for those who prefer them. It does not preclude further subdivisions within the sexes or subdivisions of mixed sex.
All this emotional energy currently going into catastrophising about what happens to the people who currently enjoy cross sex privileges would be much better spent taking off the blinders that the activists have conned people into wearing and building a new and frankly better solution from the many many better possibilities that are still entirely open.