I'm not sure how to reconcile your choice to downplay deterrents with a claim that regulation is the way forward. Regulation without disincentives is meaningless, isn't it?
To my mind, the US has a violence problem and guns much more so than a gun violence problem.
Focusing specifically on murder by guns, California and Texas, who are largely similar in murder rates are also largely similar in murder-by-gun rates despite having regulatory environments that are about as far apart as you can get in the United States. I'm not taking a position here, and I understand there might be compounding factors (like both states being border states); but it does make me wonder if what has been effective in other cultures will be effective in the United States.
I think your points about reducing poverty and providing for education are on point. Generally speaking, states that I consider "poor" states and regions within states that are generally depressed seem to have higher gun-murder rates. They also seem to have generally more violence. These are things we should focus on anyway (and would probably also help with gun-suicide and gun-related accidents).
In the US, I think both sides in this argument are generally disingenuous. Speaking very broadly... Gun control advocates manipulate statistics and call any death by gun "gun violence" as if there is no distinction between a suicide victim and a murderer. They also don't separate out legal versus illegal gun access in the use of gun murder, which I think is meaningful if you're talking about adding more laws/regulations. Some are at least honest enough to state they want to restrict all private gun ownership. Again, broadly speaking, they push laws that are about restricting legitimate gun ownership rather than promoting responsible gun ownership. On the flip side, gun control opponents oppose any change in law and while they acknowledge that there are differences in types of gun-related deaths, they aren't any more nuanced in their views about it and generally don't seem to acknowledge that non-murder gun deaths are also a problem we need to work on.
I'd much rather see everyone gather around responsible gun ownership for those that want it. (And I do think that more severe punishment would be useful in reducing accidents and maybe suicides, even though I think it wouldn't do anything for murders.)
This has turned into a bit of a ramble I suppose, but I'm just generally wondering what you think the path forward should be, why you think such a path would work in the United States specifically, and how regulations work if deterrents don't work.