have made women think that the whole objective of being trans is to be treated like a literal biological woman, and not as a woman in the context of society.
I'm afraid you are under a misapprehension. That's exactly what it means.
In the USA, two young male runners are competing against young female runners. Clearly, there's a phenomenal advantage. No hormones or surgery - nothing - and in the States, sporting success in high school can lead to scholarships to university. That is absolutely all about ignoring biology. When I've raised this with US friends - women, feminists - they've said, "but trans women are women." Shelters, prisons, rape crisis work. "Trans women are women." When I have sought to argue, I've been told that the problem is that I don't understand: trans women are women. (This also makes lesbians transphobic, if they don't want to date a trans woman with a penis. Lesbian dating apps will kick members off if they seek to specify that they only want another female. The Guardian sent a lesbian on a blind date with a trans woman, in fact. So in old money, they sent a lesbian on a blind date with a heterosexual male. But that's fine, because trans women are women?)
A rape survivor wrote to her local hospital, asking that only a female clinician performed intimate care - a mammogram. That letter was then used in training as an example of unacceptable bigotry. Yet what was asked was covered squarely by the Equality Act. That's not about accepting someone in a social role. It's about insisting that a transwoman is in every way the same as a natal one. That's where we are headed.
Most of us had no issue with social acceptance in the way you describe. Why would we? That's no different to accepting religious observance, as I said. But that's not what is being demanded now. A young man who is very active in politics in Scotland said that transwomen are women, so they should have been the only ones asked about GRA reform as they can represent women, and also trans people. He was serious.
Again: there is no issue with anyone trans living in any way that makes sense and feels right to them, and I believe and hope that the overwhelming majority of posters here support that, and would support laws to prevent harm resulting. There is a problem when women's rights and provisions, and the definitions they rely upon, are either weakened, or erased completely. It's not as if we've had them very long, is it?