Gotta disagree there. Societies are composed of men and women, speaking here purely in sexual terms. Societies then decide how to deal with them. That's gender.

I agree those slogans are awful. Sex is not assigned; gender is. Trans women are not biological women; they are men who deserve the same right to live as they please that we all should have. The problem is we're trying to talk about two different concepts using many of the same nouns and pronouns.