Sex and gender is different is all well and good. People are welcome to have gender identities if they wish. What they don't get to do it claim that having a feminine gender identity gives them access to women's single sex spaces, which were created for women in the sense of sex because of the different needs of female bodies.
It's an attempted linguistic sleight of hand and we're not buying it. If you can claim man and woman mean gender identity not sex, you can let trans identifying men into spaces that were supposed to be single sex (not gender) spaces for women. In fact, any and all men can go in, as identity is not verifiable.
For example, Gender identities don't need separate sports categories. Separate sports categories are there because of the physiological differences between sexed bodies.
I struggle to think of any provision where there is a good justification for separate provision for people with a feminine gender identity and people with a masculine gender identity. What properties do they have in common that would lead to such a need?
It wouldn't work in any case as plenty of people don't have a gender identity, or have one of the 64 other weird and colourful ones. Whereas we're all male or female by sex.