I especially take issue with this statement from the gay man: "When most people say "woman" they are refering to gender (because they aren't talking about arbitraty sex organ differences)."

Bullshit. Otherwise how and why does "misgendering" happen, and why do people have to put their pronouns up front to preemptively prevent misgendering? It is because we do in fact say women in reference to sex and not gender. Sex is real, gender is a social construct, people tend to talk about real things and social constructs are what we are forced into accepting. We deal in right or wrong, but the social construct of law is what we get hit upside the head with by those who are more concerned with the social construct than the real. We do things for each other in our communities and we help each other if we get hurt, but the social constructs of insurance and liability means a neighbor shoveling someone's driveway full of snow could end up in a legal battle if something goes wrong. We have the attributes of strength, speed, dexterity, charisma, intelligence, beauty, competitiveness, but the social construct of money replaces those things as the only currency we trade in that matters. But, we deal in the real in all these matters, and only in the social constructed versions when we are forced to. We do not refer to gender, we are not attracted to gender we do not have sex with a gender and anything beyond polite kissing involves sex organs and not gender organs. The gay man is lying.