Indeed, if the population evades the mental health care issue, it is a VERY dangerous world and perhaps your measures are not enough.
Consider gangs and home invasion fueled by drug addiction, something curable if survival instincts exist in drug addicts. In the case of incompetent mental health care enforced by the evasive public, perhaps homeowners or lessees should be allowed fully automatic weapons with maximum magazine capacity.
If we accept the necessity of regulating firearm ownership for mental illness, who gets to say whom is a danger to themselves and others? Is that not exactly how the Soviet Union crushed dissent, by diagnosing it as lunacy and institutionalizing open dissenters behind closed doors.
So is it a list of diseases, or a list of names? Who gets to decide whose name goes on the gunless list?