I agree with the parent on negotiating and I think your points are valid too. There's no one individual parameter in this discussion if you're looking to improve.
I did see something in the article you noted that I'd like to question though:
> that is, when their credentials have already been screened and they are in the interview phase—the focus shifts away from their competence and toward their social skills. That effect is absent for male candidates.
I think it's possible that this is not hiring managers trying to screw over candidates or make it harder for women, but it's a likely bias from past experiences based on a team environment where men and women differ because of our inherent differences.
Anecdotally, I've only seen women get nasty with other women to the point where it brews and stews and becomes problematic in a team environment. If women don't like men - they seem to navigate that better, perhaps because of your point in #1. Nonetheless, if men don't like other men - many times they'll blow up on each other but it doesn't linger. I've been in physical fights with great friends of mine, we've said nasty things, and in most cases the next day we let it go.
Just in the last couple years, I've had to let go several senior women because they were causing stress and anxiety with other women employees. From my experience - it's more likely I'll have a cultural issue with a female (with other females) and it's more likely I'll fire a man due to competence. Navigating cultural problems is much harder than making a decision based on lack of production. Everyone generally understands when people are fired from incompetence.
That being said - everyone has their own biases. It's not supposed to enter the picture in hiring but these considerations are often more subtle and are usually in consideration of one's current team and culture they wish to preserve. If you have some really strong personalities, male or female, you might be expressing concerns about social skills because you're trying to avoid a potential clash, not because you're out to get women. I wouldn't hire a male drill sergeant to work with my team which is more laid back and collaborative either.