Your claim made in 1) that those who support and defend female-only sports (and spaces and services) are all "right-wing culture warriors" or friends with them who take our cues from Tucker Carlson and Laura Ingraham shows how clueless you are about the people and politics involved here, and how narrow and parochial your own USA-centric POV is.
Millions of women and men around the world who oppose the incursion of males into female sports and regard gender identity ideology as sexist, male supremacist and harmful have always been progressives who've worked hard to promote tolerance, diversity and inclusion of minorities and those who are marginalized.
Women like Martina Navratilova, NancyHogshead-Makar, Mara Yamaguchi, Sharron Davies, Chris Evert, JK Rowling, Cathy Devine, Linda Blade, Kathleeen Stock, Julie Bindel, Kara Dansky, Doriane Lambelet Coleman, Maya Forstater, Janice Raymond and men like Ross Tucker, Daley Thompson and Jon Pike hardly are being driven and exploited by American right-wingers like Carlson, Ingraham and Matt Walsh.
In particular, women like me approach the issues at hand here from a longstanding second-wave feminist perspective that has always been critical of the sex stereotypes that constitute gender. We see the sex stereotypes that make up gender as means that inculcate and perpetuate the traditional sex hierarchy in which males are regarded as naturally superior beings meant to dominate and rule over females, and females are seen as second-rate and naturally second-class beings who embrace and enjoy our inferior social position. Gender teaches that the wants, needs and demands of males must always come first, and females must always put our own interests, wellbeing and safety aside to make sure males get what they want.
In short, the "gender critical feminist" POV and is entirely different to the perspective of the right wing.
As for your claim in 2): many who most staunchly oppose the incursion of males into female sports and spaces in the name of gender identity ideology are ourselves people who have spent our lives rejecting and not conforming to the sexist stereotypes that constitute gender and which today's gender ideology promotes. Many are lesbian, gay or bi too.
Some of the most vocal defenders of female-only sports, spaces and services today are left-wing butch lesbians like Joey Brite, Linda Bellos and Allison Bailey and gay, bi and het men who were and are notable "gender benders." Including Marilyn Peter Robinson, the popular gay British singer known as Marilyn in the 1980s who now goes by the handle Mister Marilyn on social media. Marilyn, or Maz as he is also known, is still an active "gender bender" today in his 50s, but because he says humans can't change sex and he sticks up for the rights of women and girls, he is widely lambasted as a "transphobe" and right-winger.
Moreover, many people today of both sexes who identify as trans, non-binary, gender fluid, neutrois, maverique and so on who nonetheless oppose males using their claimed gender identities as an excuse to horn in on female sports, spaces and services. They see that there is a massive conflict between what gender identity ideologues are demanding in the name of "trans rights" and the hard-won rights that the female half of the population in the Western world only obtained relatively recently in history - and only because generations of women fought tooth and nail for them.
Just as the behaviors and attitudes displayed by XY DSD athletes like Caster Semenya have not helped to advance the interests of people with DSDs generally, those who advocate intrusion of males into female sports, spaces and services and who preach human sex-denialism in the name of "diversity, equity and inclusion" are really not helping the vast majority of people marginalized for not conforming to traditional gender paradigms.
In fact, many of us believe that the gender identity ideology that's being advanced today in the guise of tolerating difference and being progressive is at its core as regressive, conformist and male supremacist as the traditionalist views of right wingers like Tucker Carlson and Matt Walsh.