Well, let's talk about the types of gun violence, and what MOST Americans are most concerned about (listing in no particular order):
Type 1: criminal-on-criminal violence. This is the most common way to be the victim of third-party gun violence. Drug deal gone bad, gang revenge killing, mob hit, all that stuff. Yes, there are occasions when the victims are innocent (murder during a mugging, etc.), but that is a very low percentage. Almost all of these cases of gun violence are a direct result of being involved in other criminal activity. So, MOST Americans realize that they can avoid this type of gun violence by not engaging in other criminal activity. And yes, there is some bystander risk (see some victims on 6th street), but the numbers just aren't that large. So, I'd presume that most of us around here aren't terribly worried about this one, individually (aside from our bigger concerns about overall crime and such, and sure, the social risk of bystander harm, which is horrifying when it happens).
Type 2: suicide. This is one of the leading causes of gun death, and almost always happens with a gun already in the home. In my mind, this is perhaps the #1 reason that guns should be locked in a safe, with only one (stable) person with the key. In my home, that person is me (yeah, stable is debatable). But that's definitely because I have kids who are young, emotional, and have even struggled with depression at times. We should be more concerned about this than we are -- it's low-hanging fruit, and offers a great ROI.
Type 3: accidents (either in the home or in the field). For "at home," see the para above, that's the best way. As for "in the field," good safety training and practices goes a long way. Texas Hunter Safety covers it. SOME accidents will still happen, nothing's perfect. But this is another low-hanging fruit thing: keep your guns locked up, require gun safety courses. Most people aren't too worried about this one, but they should be more worried than we are. If we placed a greater emphasis on gun safety, that would certainly prevent a good number of needless deaths.
Type 4: domestic/spouse/acquaintance violence. Husband shoots wife, jealous lover shoots rival, two friends have a beef, that sort of thing. There's some "sudden passion" risk in here (so, if someone didn't have immediate access to a gun, maybe a shooting would be prevented). In many cases, these are known risks (history of domestic violence). Americans do have some general concern about this, because many of us know first or secondhand someone who has been a victim of such violence. In any case, these are situations where the victim knows the shooter well, and there is usually some personal history behind it. This does involve mental health some -- in fact, most "mental health" violence is in the home/among family and such (a fair amount of domestic violence has overlap with mental illness of the bad actor). There are many mechanisms that could be in play to reduce this risk (Domestic Violence Orders, etc.).
Type 5: Random shootings. These are what most people mean when they say "mass shooting" -- Uvalide, El Paso Wal-Mart, Buffalo, Tree of Life Synagogue, Parkland, workplace shootings, etc. And these actually scare Americans quite a bit precisely because they're functionally random, and can't be avoided by actions the victim would take -- like not being involved in other crimes, keeping their guns locked away safely, etc. It's the terrifying element of the "bolt out of the blue" nature of them, as well as the high number of casualties each incident can involve. Whether it is driven by racial hatred (Buffalo, El Paso), actual mental illness, broken young people with not-yet-developed brains who are incapable of dealing with strong and frustrating emotions (likely Parkland and Uvalde, also people who are broken socially (think "incels")), people who are incapable of dealing with workplace disappointment, we see these as being beyond our control. We can't control when some shitheel just snaps, or at least it feels like we can't (we actually CAN do things that help stop these things, like good HR training to spot warning signs, monitoring students for troubling behavior and, oh, I don't know, we could stop promulgating bullshit racial animus ideas like "replacement theory" that are destined to foment stochastic terrorism).
I'm sure we could come up with another type or two, but this seems to cover 95% of gun violence. So, the questions are -- if we want to reduce gun violence and gun deaths -- what measures can we reasonably take to address any, some, or all of these categories? Talking about a measure that is mostly aimed at one Type as a solution to another Type may not make sense. Or, it may help a good bit for a couple of types (e.g., if there are guns in the home, having them locked away with the key hidden or combo known only to an adult will prevent both suicides and an angry kid from getting dad's gun and going on a rampage). But we also need to account for what actually concerns people the most. For all the "but Chicago!" rejoinders, it's not an effective one because most people don't give much of a shit if two dirtbags off each other in a gang shooting. Concern is about risk and control. My family is wildly unlikely to ever be victimized by Chicago gang shootings. They are more likely to be the victims of random gun violence. They are also more likely (because our household has guns) to be victims of an accident or suicide. So, our focus is on the things we can control, and are reasonably concerned about. Understanding these things, we can then have a more targeted and rational discussion about gun policy.