So my opinion is that there are broadly three facets of what people call "gender".
The first is sex, which is set at conception, is immutable, and is likely to remain immutable for at least the lifespan of everyone alive today and their grandchildren. It means that women are always and unavoidably saddled with childbearing and men are almost always bigger and stronger than women, all things (training, experience, etc.) being equal.
The second is gendered behaviour, things like the "people/things preference" distribution etc., which seems to be somewhat inherent and driven by our evolutionary past, and distributes such that females are more likely to have "feminine" behaviours and males more likely to have "masculine" behaviours. But men and women are far more alike than they are different, and the most "masculine" (according to behaviours) person in a given space could easily be female, and the most "feminine" could easily be male. And there is absolutely nothing wrong or even surprising about that. The best person I ever worked with when I worked in elderly care many years ago was male, and the best mechanic I know and trust is female.
The third is gendered social expectations, which are entirely constructed, are usually a bastardisation of observations on gendered behaviours, and are almost always harmful to, if not the majority, then a very sizeable minority of people. I also think these are increasingly set by the influence of large corporations and driven by corporate interests (eg: "let's make pink Legos so little Anna can't inherit her big brother's set").
Gender ideology is especially dangerous because it uses the third thing (socially constructed gender expectations) to foster insecurity in young people about the second thing (a completely normal distribution of masculine/feminine traits across both sexes) and suggests that this may mean that the first thing (immutable sex) is wrong. Couple that with the monetary interests of clinicians who can make lifelong patients out of young people and the political interest of those who see gender "queering" as a way to break down the current order of society, mix in a little "toxic masculinity" rhetoric with feminine and sensitive young boys, a spoonful of abjectly pornified, instagram-filtered femininity with bookish, masculine young women, a dash of internalised homophobia, and you have a recipe for absolute disaster.
That's my take on the whole thing, anyway.
I don't want men to have access to women's single-sex spaces. I also don't want feminine young men or masculine young women to feel like there is something "wrong" with them, even if they want to buck socially constructed expectations about appearance.
My own mother's answer to me hating dresses, long hair, and dolls was to put me in dungarees, cut my hair off, buy me toy cars and magazines about bugs and let me roll in the mud.
I think she was onto something.