It might be helpful to provide some Australian context.
The Australian Sex Discrimination Act was amended in 2013 to remove the definition of "man" and "woman" as a biological distinction, and "gender identity" was included in such a way that anyone with female as their gender identity can now be considered, in law, to be female. The impact of this, as it is currently being interpreted by the High, and the Supreme court, is that biological women are no different to men who claim a female gender identity. In short, we have no legal grounds for sex-segregated spaces, and any that remain are open to a legal challenge that will be won. Some people celebrate this as progress, but there are many of us who understand that this has a detrimental impact on wmen and on girls.
City Councils in Australia are responsible for a broad range of local services and facilities, including parks and their public toilets, public swimming pools, libraries, public building hire, and local by-laws such as protest permits and such like. The Deputy Mayor is the 2IC of the elected council. The Mayor and the Deputy Mayor are typically, and in this case NSW is typical, voted in by the council, rather than directly by the public.
In a city the size of Sydney, this gives the Mayor and the Deputy Mayor considerable influence over services and facilities relied on by many women and girls in their day to day lives. Public swimming pool and public toilet facilities, local libraries where mums and babies and toddlers go for weekly story group time etc. Councils also provide grants for local community groups, sports clubs etc. Womens groups must apply to council for the Australian version of the "let women speak" gatherings and such.
It was in this context that my post questioned, well tbh, assumed that considerations of the needs of women and girls to have legal sex-based boundaries in place are under further threat when the Deputy Mayor, a person with considerable power and influence, shares the ideology that men can be women, and therefore should have access to spaces previously for women only.
Watching the legal environment in Australia deteriorate for women in this way has made me angry and outspoken, and I can see by some of the commentary on here that angry and outspoken women with sex-realist views must tone-police themselves, lest thay come across as angry and outspoken, and give those who are stripping away our rights in the face of a largely silent electorate, ammunition to say, well, they are angry.
When a male person claims a female identity, I do make some assumptions. I will assume that they wish to support the trans community in their desire to access female spaces. With no laws to otherwise protect those spaces in Australia, I see this as another step backwards for womens rights.