Sure, I'll just go through each of your points individually.
"The variation among men and women between the Big Five personality traits has been proven consistently. A study from Stanford notes differences among the hippocampus and amygdala (they do stress this does not make one gender more competent than another)."
I don't disagree with this actually. My argument in the article isn't that there aren't observable differences, it's that there aren't INNATE or universal differences that we've been able to prove. I'm fully aware that there are personality differences between men and women in our society, and that brain differences can emerge. But the evidence suggests that these differences are created by culture rather than being biologically inevitable. That explains why the gender differences in our society aren't present in all cultures around the world.
And like I explained in the article, taxi drivers develop a bigger hippocampus over time because of their lived experience in the world, not their genes.
"Similar studies have noted that as gender equality is pushed in various first-world countries, the gap between men and women in certain fields grow. For example, in Scandinavian countries (where they have gone further the anyone in the strive for gender equality) the ratio of men to women is 20:1 engineers vs 20:1 women to men for nurses."
Sure, but how does that it any way prove that women are biologically more prone to being interested in those fields? Do you think culture doesn't exist in Scandinavian countries? I myself pursued a more "feminine" field, anthropology, and I'm fully aware that the "feminine" values that were instilled in me throughout my life made me more likely to become interested in studying humanity. If a country is more equal, it doesn't mean that there are no longer gender norms that affect people's interests and behaviors. I'm not even saying those gender norms are all bad. I'm just saying it's scientifically inaccurate to attribute them all to biology.
"There is even research between young boys’ interest towards “thing” based toys while young girls’ were predisposed to go towards “people” based toys. This is at an extremely young age, and would not be due to any “social programming.”
The studies like this that I've seen have been affected by issues similar to what I describe in point #5 of the article, where I discuss poor study design. But even if a study did prove those differences in preferences, where is your proof that children that age are immune to learning their preferences from social cues/their environment? Babies are extremely sensitive to cues about how they should behave the second they pop out of the womb. If a parent even does something as subtle as smiling in response to a behavior, it can reinforce that behavior. The vast majority of parents teach their kids, explicitly or not, to play with the toys that correspond with their assigned gender. Boys' preferences to play with trucks doesn't prove any innate difference. I loved playing with trucks. My brother enjoyed playing with trucks AND dolls but that desire was quickly beaten out of him.