I'm not sure if I can bring anything helpful to the discussion, but in a sense I may have a somewhat unique perspective on this amongst the people in this thread.
I've lived in places where legal gun ownership is possible but much more strictly regulated (Germany), others where it's more or less outlawed (UK), my mum lives in a place where every (male) citizen under a certain age has a full on military assault rifle in their wardrobe and knows how to use it (Switzerland). And now I obviously live in the US, and have mostly lived in places where hunting is part of the culture and where people in generally own guns. And yes, we own a couple of long guns, although we probably need to get them checked over as they haven't been used in a while. Plus, both my wife and I are pretty decent shots and I do have an interest in mostly older firearms.
Germany didn't really have that much in the way of strong gun regulation until the 70s, until the domestic terrorism decades (basically the 70s and 80s). Of course said terrorists mostly stole the guns or got them from "supporters". You know, like the Stasi. But anyway, these days you have to have a reason to own a gun (like for hunting or sport shooting), have to get a proper permit and have to have secure storage. Doesn't prevent you from owning, collecting and shooting guns, but like everything in Germany, you have to deal with the bureaucracy. Gun crime is relatively rare (and thus headline worthy) and usually involves guns that were either illegal and/or stolen in the first place, including stolen from the police. Then again, this is a country where you can get sued from calling someone an a*hole, so it's a slightly different vibe overall, and yes, it's mostly the outlaws and the cops that have guns. And the odd old guy who has a WWII tank in their basement although rumour has it that a fair number of families probably "forgot" to tell the cops about gramp's WWII souvenirs back in the 70s.
Switzerland is probably closest to the US when it comes to European gun laws. In some Kantons, you could (probably still can) walk into a gun store and walk out with an AK47 with very little in the way of background checks. Heck, a bunch of enterprising people had a pretty good pipeline going from Switzerland to ex-Yugoslavia back during that civil war. Then again, in a fair amount of places the attitude is "heck, everybody's already got at least one of these in the house anyway", although my mum did mention that not too long ago, regulations were changed and while you're military reservist, you keep the guns at your home, but you don't get any ammo anymore, mostly due to the number of suicides that occurred. Again, gun crime is fairly low but does happen.
The UK banned pretty much any kind of private gun ownership following the Dunblane school shooting and unless you're a farmer, it's pretty much impossible to legally own a functioning firearm. They also have very strict laws for when you get caught with either guns or ammo, mostly left over from the heady days of the IRA and loyalist paramilitaries (although those were mostly overlooked, it was really the IRA that was targeted by these laws). Yet, guns aren't exactly in short supply in criminal circles (because you pretty much have to be a well 'ard crim to risk owning one) and it's not that uncommon in proper gang infested areas to have shootings. But most of the impromptu disputes are settled either with fists or knives. Also, the coppers in general aren't armed, but the firearms officers can be a bit of a trigger happy bunch and have been known to blow people away for fare evasion or carrying a table leg.
Out of the three, I would argue that the UK has probably the most "low level" physical violence, and societal acceptance of it (at least in working class circles). This goes back quite a while, but they can't really ship the troublemakers off to the colonies anymore. Out of the three countries I mentioned, it's also the country with the worst mental health support out of the three and despite knowing a few people who have been saved by it, not necessarily a system you want to get caught up it. Oh, and it's also the country that's probably closest from a societal perspective when compared to the US (large and increasing wealth gap, a very depressed and shrinking manufacturing sector and a fairly limited social safety net).
What does any of this have to do with the US? Well, to me it shows that reasonable regulation can work, as long it strikes the right sort of balance. The UK IMHO went too far, and the main result seems to be that hospitals in certain areas like Glasgow now have world class experts in dealing with knife wounds, and that's despite strict knife laws that can even get you into trouble if you're a chef and walk home with your tools.
I personally don't think it's unreasonable to expect people to submit to a background check before buying a gun, and maybe even have to verify (to their insurance if not a regulation authority) that they have safe storage for a gun. I also think that with the number of guns floating around, it's probably unrealistic to expect a supply reduction, but some more sensible policies around transfers/personal sales (that protect both buyer and seller, especially if it turns out that you unintentionally ended up with a firearm you bought privately that, say, law enforcement may have an interest in) would probably help.
To a certain extent, I also find the invocation of "mental health" in this context problematic. Yes, I think it does play a role, as does having a closer look as a society into the, well, mental health reasons that are contributing factors here. But I see it being used in the context of Minority Report like "pre-crime" approaches way too often, and given that law enforcement these days seems to be at the front line of handling mental health issues because other institutions have been gutted/defunded, I think that's a very problematic approach.
It's become too easy to use it as the hammer on nails that stand out for being slightly different, and if anything might end up doing more harm to people who are not 100% conforming to an externally imposed idea of what normal is. If anything, stigmatizing not living up to other people's ideals isn't going to make this any better and very likely worse.
All that said, I also think that implicitly putting more firearms out in the wild by making it easier for people to open carry (and more importantly, concealed carry with fewer restrictions) might not quite be a step in the right direction here either. But without addressing the reasons why people feel a need to carry, I don't see that flying much either.
Anyway, mostly a perspective of someone who grew up in a different culture around firearms that might or might not be helpful to this discussion.
