Partially correct but you are conflating the movement fighting for biological rights (eg: reproductive rights) as it being bio-essentialist. And there certainly was infighting about trans people within second-wave feminism (eg: feminist sex wars), but then there's also intersex people. Second-wave feminists more generally did not have the kind of one-drop rule towards womanhood as you do, where someone could have lived their entire life as a woman, be perceived as a woman, experienced misogyny as a woman, experience patriarchy as a woman, suffered domestic abuse as a woman, have breasts and a vulva, etc, but once some test determines them to be intersex, you disqualify them from womanhood entirely and cast them as male. Second-wave feminists would not have done this. In fact, I believe even Greer deplored surgeries being performed on infants to make them comply with society's perception of the binary.