My view is that guns are awesome toys, maybe one of the most fun toys a grown up can play with. Most people who I know who own guns own them because of this fact. Usually they also mutter something about personal safety or protection from tyranny, but they don't have very developed ideas on what this means when you press them further. In recognition of that, it's only natural that people would want bigger and better guns that can load faster and destroy more things quickly.
So it's my feeling that people should be able to play with these toys in designated spaces like gun ranges, etc. They should be licensed to do so and their higher capacity weapons should be stored on site. Gun range/storage facilities should be granted non-profit status and should not be allowed to also sell or buy firearms. A strict firewall should be kept between them as a licensing, storage, and usage facility and the sale of firearms.
This leaves us with the actual issues of self defense and hunting, neither of which require high capacity firearms. Simple shotguns and hunting rifles should be allowed in the home with state level (rather than federal) licensing. There shouldn't be mental health screening because I tend to find people are more likely to grow up with guns already in their lives before they know about their own mental health and so adding this requirement would disincentivize people who need it from seeking treatment. Instead licensing should focus on safe handling and criminal history.
At the same time, federal monies should be committed to establishing and/or expanding mental health services in high gun ownership communities as well as communities with high incidences of gun violence. It won't be a silver bullet, if you pardon the bad pun, but it will help.
This only leaves us with the problem of handguns, which I believe should be eliminated from society. Cops shouldn't have them and neither should private citizens. An aggressive buyback program followed by a ban should help, but it would be a challenge to stop their illegal flow, as we've seen in the UK. I honestly don't know how this would be accomplished, but all of the people I know who have died in firearm related circumstances have died via handgun, and that number is much higher than I wish it were. Obviously there needs to be more emphasis on the social problems that lead people to pick up a gun, but handguns being so easy to get is an indicator of their easy flow through the market. Giving them biometric safeties is one way to curb some of their circulation, but interventions like that would only work in an already recessed gun market. If people still wanted to play with them, they could also go to the storage lockers, but would be quarantined from home use.
Of course the problem with any central location for weapons is the risk of theft, but armories have existed for centuries without somehow having to repel attackers all that regularly.
I've proposed a similar schema on here before only to be told that the 2nd amendment makes unmolested ownership uncontestable, but that's led to some quibbles over the intentions of the framers of the Constitution, who were responding to specific historical circumstances. Another answer to that challenge is that we today have many caveats to how much we can exercise the far more important first amendment. Freedoms of speech, press, assembly, and certainly to redress grievances to government have so many limitations on how, when, and where we may exercise them that we just accept and often encourage. For some reason, this same standard isn't applied by 2A fundamentalists, who only grudgingly concede to the restrictions on gun type ownership or open display now.