There are various angles and layers to the 'trans movement'. Some parts are certainly spearheaded by women or enabled due to the support from women.
The ideological frameworks in academic feminism have given legitimacy to many of the arguments put forward by queer theory, gender studies etc. I don't think feminism is unique in thinking that sex differences magically only take place below the neck, or that evolutionary impacts on human behaviour can be ignored. It seems like many sociology fields have similar approaches and think much or all of human behaviour is socialised or socially constructed.
Universities and academic institutions churn out a certain uniformity of thought and people who have been trained in this type of mindset then go on to various public sector roles that influence policies and practices.
I think it's easier for people to believe there are no differences between women and men or that any physical differences don't matter if you are in fairly white collar, office based jobs - there's a reason why mainstream feminism seems to always campaign for more women CEOs rather than more women oil rig workers.
I have a vague recollection of a study or research that showed that as more women employees are hired in a company, the acceptance of minority groups within the company also increases i.e. as more women are hired it reduces overall sexism /sexist practices but also leads to better working environments for gays and lesbians for example. I don't know if the research was widely supported but it was used as evidence to encourage more women being hired in the workplace to balance the stereotypical image of male dominated businesses.
Does this have the inadvertent effect of trans policies being more quickly adopted as it is another minority group that is embraced?
Women are, on average, more agreeable than men and likely to accept or put up with situations and policies even if it might not be good for them. Apart from the queer theory adherents, most women who are supportive of trans inclusive policies are under the illusion that transwomen are a delicate, fragile minority that have had genital surgery etc. Trans activists deliberately don't correct this public perception because they know their policies will only gain acceptance if this illusion is maintained.
There are lots of other influential layers to the trans movement, such as medical/surgical establishments relishing playing god or the pharmaceutical industry with a vested interest in promoting certain ideological narratives.
Women in positions of power and influence are not immune to thinking they can achieve fame for supporting 'the next big thing since gay rights'. I think a lot of women in public roles will eventually end up looking very foolish and silly for their stances. The men who prop up the trans lobby generally prefer to let women be the public face and therefore let women take the inevitable fallout from publicly committing to an ideology. I mean, women like Lisa Nandy and Dawn Butler will continue to be remembered and laughed at for their views of rapists being in women's prisons or humans being born without a sex. The men who have supported gender identity ideas will manage to salvage their careers and keep their reputations intact and get a free pass, while the women will be viewed as incapable of being trusted with public policy.