"Regulation" per se doesn't actually solve the underlying problems, it just masks it. Solutions are things that address the underlying issue. In that case of most violent crime, that's poverty, systemic racism, lack of opportunity, economic inequality, and so on. All of those things would also tend to decrease mental health issues, but going farther and making sure that people could realistically access good mental health help--without fear of loss of rights--would improve outcomes.

Moreover, in the example I used, there were three causes of crashes; there are absolutely, definitely cases of people using a car to intentionally kill pedestrians in indiscriminate attacks. Regulation doesn't prevent that kind of abuse, unless you want to simply, y'know, entirely ban the use of all vehicles. Which would require entirely overhauling all of the infrastructure in the US.

As far as 'school shootings'--which I'm defining as violence targeting schools with the intent of causing mass casualties--you don't have a single solid cause. There's a report--by the US Secret Service, I think?--that tried to nail down factors, and what they came up with was that there wasn't any single thing that most or all of them had in common that wasn't a legal and protected activity, or wasn't common enough that targeting that factor wouldn't have widespread unintended consequences. E.g., if 50% of the students that committed mass murders were bullied, then putting all bullied kids on a watch list would be punishing an enormous number of kids for the actions of a very tiny minority.