"Man" and "woman" could indeed change meaning from the current, almost universally understood "adult human male" and "adult human female" to be more akin to one's gender, a social construct largely based around stereotypes.
If that is going to happen, there is still a need for female and male sex-segregated safe spaces. And I have a feeling it isn't enough for many transactivists to be legally their acquired gender; they will still want access to the sex-segregated facilities designed for the opposite sex.
Stating that one sex class places another sex class at risk through violence, sexual abuse etc is not transphobic, it is an objective fact. Stating that, post-transition, trans-identified males still commit crimes at the rate of other males is not transphobic either. Stating that the reason why females are almost universally the oppressed sex, across geographical, cultural and religious boundaries, is due to their snaller size and ability to bear children is not transphobic, it is a fact.
I do not deny anyone's right to exist. I do not deny anyone's human rights. I defend my own sex class's right to safety, which at times requires sex segregation.
The irony is, trans-identified males have used women's facilities for years. The aggressive and homophobic trans movement we see at the moment is at least as damaging to them as it is to women.
Sadly, removing gatekeeping and enshrining this in law only endangers people; it will not make bigoted and violent men any more accepting of transpeople.