You have misunderstood the position entirely.
I don't have an "ideological" belief that society "should" be segregated, nor laws defined, based on some abstract belief about ideology.
Actually I would much prefer that neither sex nor gender segregation were necessary.
But Women (in the female meaning of the word) have historically been socially, poltically and culturally marginalised and abused by men because of their bodies. That is an undeniable fact and changing the meaning of the word Woman now does not change what happened to those people, and why, in the past, nor its continued existence in the present.
And Women (in the female meaning of the word) have historically been financially, culturally and poltically disadvantaged because of the asymmetrical demands of reproduction and child rearing on women vs men, leaving women with fewer resources and less time to put into other activities, and compounded by the vicious circle that the more men accrued poltical, financial and social advantages, the more the standards and institutions of society were set up to accomodate male practical needs and socially accepted behavioural norms and not female ones. That is an undeniable fact and changing the meaning of the word Woman now does not change what happened to those people, and why, in the past, nor its continued impact in the present.
And these facts conspire to mean that Women (in the female meaning of the word) face risks and limitations that men (in the male sense of the word) do not, and will continue to do so until the unconscious biases that are built in the foundations of our society and culture are eradicated.
So while ideologically I may believe segregation by both sex and gender to be a bad thing, practically I believe segregating by sex is currently a necessary evil, and will remain so until society no longer disadvantages and abuses female people physically or socially.
Furthermore, I see a deep logistical inconsistency in the genderist assertion that everything we understood a woman to be based on sex is wrong, and yet everything social and legal built on that wrong basis is somehow just right, the only problem is that it should have been determined by an inner, undetectable, self-defined gender identity not the externally perceived fact of sex. I cannot see how, had we not already had those sex-based segregation, gender identity alone would have lead to them and therefore, whether or not gender identity is empirically real or not, I cannot accept the claims that are being made in its name.
It is this, not the existence or not of gender itself, that has lead me to conclude that genderist demands for social change are ideological not based in any practical understanding of womanhood or the reasons for existing sex based provisions.
I would (and have said in other threads) be not only supportive but quite excited by a social movement to recognise similar personalities, regardless of thrir sex, as meaningful groups who can legitimately lobby poltically and socially for their own needs and in some cases for that to be more relevant than sex. I think it could even be a path to a point where sex genuinely is less significant in life outcomes rather than what the genderists currently want, which is no less significant but just ignored.
What I cannot accept is the demand that sex is simply replaced by gender with everything kept the same, because despite the genderists' ideologically driven willful blindness to hearing female voices, being female in our society has consequences.