Well yea, if perception of chess skill corresponds with actual chess skill then it means people are capable of self evaluating chess skill.

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?