All that said. it would eliminate those who have longer term issues, a history of violence, violent criminal record, etc.
You're already restricted in what you can buy, carry and transport.
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.
Haven't we already proven that the US is vastly different from Canada? and not just in gun laws but demographics, population, and values.