Is there a reason why that viewpoint is out of line or anti-gun?
No, it's a valid question, but you're not going to like the answer.
Many of the very people you're hoping to filter out are the most adept at acing those sorts of tests. I've met a sociopath. They do a great job of appearing normal, happy, and friendly. He'd get a gun.
Meanwhile, a guy who listens to Nick Drake, reads Rod McKuen, and gets in a bit of a funk sometimes gets denied for borderline depression or whatever. Despite being an empathetic, moral person, he doesn't get a gun.
Then there's the question of who defines the criteria and limits. What exactly defines "too depressed" to pass? Is there an appeal for denials?
It simply won't do what's intended, and it will deny guns to folks from whom we've got nothing to fear.