Language does matter. Stonewall, Mermaids, Gendered Intelligence, GIRES, Action on Trans Health and so on have all worked hard on trying to get the public - and more importantly, government, councils, schools, charities, big businesses and the NHS - to accept that transwomen are women, transmen are men (oddly, always in that order
).
You can only accept that slogan if you are prepared to accept that biology doesn't matter as much as gender identity. (I've never seen a clear definition of gender identity that doesn't involve gender stereotypes. When I was growing up we were trying to overcome stereotyping, now we're back to saying it's innate. So much for progress.)
If you do accept it, though, what you are accepting is that there's a group called women, which can be subdivided into transwomen and ciswomen. Because TWAW, and because there must be no gatekeeping, just immediate and total acceptance of anyone who says they're trans, you also have to accept all of the following.
- Male offenders, including sex offenders, who tell the police/prison service they identify as women will have their crimes recorded as committed by females.
- Same group may end up housed in women's prisons.
- Girl Guiding won't tell parents that their girls may be sharing accommodation with a transgirl or that the Leader is a transwoman.
- Transwomen are entitled to compete against women and girls in sport, even though hormone therapy and even sexual reassignment surgery (which most TW don't have) will leave the TW with most of the advantages of the male physique.
None of these things would be happening without the initial push to get transwomen accepted as a subset of 'woman' instead of 'man'.