[ClydeFrog]

To me there are a few ways govt agencies could gauge if a person is competent or stable to buy-carry firearms. Some are already in place others could be added;
If they are under 18 years old(depending on local hunting laws or target use). If a illegal alien or fugitive/pending criminal charges/convictions of violent crime or felonys. Persons with less than honorable discharges or general under honorable conditions should not be allowed to buy or carry guns. Persons with active trespass notices or bar/not allowed on property in the state or jurisdiction where they reside should be allowed to buy guns or ammunition. Persons with PFAs/restraining orders etc filed against them(like the federal Lautenberg Law). Persons who get any documented disability, aid or pension related to a mental health disorder or inability to maintain gainful employment due to a stress/social disorder(PTSD/ADD/clinical depression/suicide attempts/involitary commitment/etc). This includes military/combat veterans or public service employees who retire-separate due to mental health issues. People who can not safely load, unload, operate, or carry a firearm due to injury or medical condition. LE agencies must separate recruits or applicants all the time because they can't safely fire or function with a loaded firearm. It happens often. People or students who apply for state or public education and are declined due to severe mental disorders or unstable conduct(threats, fights, expulsions due to safety related violations, harassment-stalking).
These are a few public laws or policy issues that could come up. These are just examples & are subject to modification due to HIPPA laws, court rulings, etc.

Dr Drew Pensky(Dr Drew) made a great, 2A supporting point a few weeks ago that many unstable & dangerous people buy guns or get weapons only because of the stupid restrictions & legal hassles they put on mental health professionals.
Those loop-holes need more close inspection than gun bans or anti gun laws.

Clyde