How about a background check and a mental check? Gun owners always call it a mental health crisis, so let's make sure all new gun owners are vetted and have to pass a psychological exam.
That would pretty much guarantee that people who need mental health services would deliberately avoid getting it to prevent it from showing up on their records, e.g. a security guard who needs to carry a firearm for their job.
Uh, what do you think a psychiatric exam is? There's no crazy-o-meter that they stick into an orifice to objectively determine whether you're mentally ill or not. You simply get asked a series of questions which you can reply to in any way you like.
"socially awkward people aren't allowed to have guns" as a policy would be a flagrant violation of a constitutional right.
The scary thing is that mentally ill people can intentionally mask their illness in order to pass exams.