I used to fly so I know all about those as well, but say you take your required recerts and then 2 months later decide to crash your plane into a building. There is no way to predict that, even with a psych eval. If you snap in between those required timeframes, you can't blame the system.
All that said. it would eliminate those who have longer term issues, a history of violence, violent criminal record, etc.
If they want me to go to a class every couple of years, I'll do it, but don't try and restrict what I want to buy or limit my options.
You're already restricted in what you can buy, carry and transport.
It will just inflate the black market.
That's the point, it certainly appears to be the exact opposite. It hasn't completely eliminated the black market in other countries, but it has made it much harder. Hard enough that the rate at which guns are used in crimes is significantly less than in the US. Can you buy an illegal gun in Canada, Japan, England? Yes. Is it easy? NO. What's easier? Go to the US, buy a gun and smuggle it back. Not too many drunk ex's and wannabe gangbangers do it.