That logic boils down to a very basic refusal to accept that trans people are the gender that they say they are rather than the one that they were assigned at birth. For the self-styled “feminists” who follow it, this manifests in a refusal to accept that trans women are women, and thus that their presence in “women’s spaces” such as changing rooms, rape crisis shelters, and gender segregated hospital wards, is not a threat.
Yes, we refuse to accept a claim (that male people can become female people, and solely by saying, "I am a woman" and immediately pose no risk to women at all) when that claim is contradicted by the actual data on crime rates. Male people are a fucking threat; pretending otherwise just puts women in harm's way. Either you care that male people (nothing to do with trans; trans people are no more likely to harm anyone than anyone else of their biology) are a risk to female people, and therefore we need single sex spaces, or you don't.
It's also very tiresome to have the accusation of transphobia flung around, each and every time women point out that no, we don't identify as a doormat. If sex weren't under threat as a criteria, and women weren't having their right to sex-based provision threatened, there'd be no dispute at all, other than the quality of care given to gender questioning young people needing to be infinitely better (which is hardly an issue being raised solely by us).
I do like the suggestion that modern gender theory is akin to homeopathy, though. Very apt; I could not agree more.