Let's say I want to measure ability in chess. I give everyone a self report survey: how good are you at chess, 1 to 10? Group A averages a 6, Group B averages a 5. Reproduced across time and regions. Is group a better at chess? We have no idea. All we know is that they report a high average than group b.
Switch to big five: "I feel comfortable around people." Rate the accuracy. Even if group a rates more accurate than group b, what do we really known from this? Comfortable is a relative term. Do group a and b have different perceptions or different experiences?
It's same for Big Five. Self-reported neuroticism correlates with, say, diagnosis of depression. It's not just perception because it correlates with outcomes of interest.
But that's not your example. If your example was self reported depression correlates with clinically diagnosed depression, we now know that people can self-perceive depression the same as clinicians diagnose it.
Your example is self-reported neuroticism corresponds with depression. So what? How are we any closer to knowing if we measured neuroticism? Is there a pre-existing link between neuroticism and depression? How did they measure neuroticism to establish that link?
What are these well known and scientifically established differences in the distribution of traits between men and women?
The research linked in the the document are based on The Big Five Personality Traits: https://en.wikipedia.org/wiki/Big_Five_personality_traits
These traits are measured with self-report surveys. I.e. personality quizzes. Not to malign the field of psychology, but this isn't the gold standard of science, IMHO.
Having read Mr. Damore's document, I can find very little in it of substance. It deals with politics and gender in only the very broadest of terms. It does not delve deeply into the scientific issues at hand, instead sprinkling his platitudes with wikipedia links to give them weight.
That being said, I completely agree that the US's nasty brand of liberalism is in a sad state. Our response to this manifesto should have been a cold critique of its many flaws, not the blind hysteria we've apparently opted for instead.