I’m starting to think the biggest issue is that people do not understand the difference between the word gender and sex.
Yes, that is the fault line in this whole debate, and it's a big reason why feminists are having difficulty making headway in our arguments.
When taking this argument to politicians, journalists, any policy makers, it's essential that we drag the argument back to the ground of sex as a material reality which cannot be changed, and which must be socially and legally recognised as such.
The framing of the debate as being about 'competing rights' between 'women and trans women' hobbles us from the start, because it's essentially a category error: it is not about competing rights between two groups, but about the attempt by one group (trans males) to extinguish another group (females) as a socially and legally meaningful category of persons.
The creation of 'gender identity' as a legal concept by its nature eliminates 'sex' as one, since gender identity redefines sex as a feeling-state instead of a biological reality.
Not having a word to delineate us by sex means all our rights (that are not general all purpose human rights) are literally erased. Accommodation of gender-based self-identified TIM folk into womanhood = ejection of those who are materially women by dint of their sex.
Very well put. Instead of getting bogged down in arguments about who suffers more, or how likely trans-identified males are to assault women, etc., we need to keep insisting on the objective reality of sex over the entirely subjective concept of 'gender identity'.