I think the idea is you want to make it as difficult as possible for mass shootings to happen. Background checks are a must, and any seller that doesn't do it properly has just as much blood on their hands as the mass shooter. if you restrict the ammunition a private citizen is allowed to have at one time, you're less likely to have mass shootings. I'm not entirely sure how Red flag laws would work in practice, but it seems pretty dangerous to have family members apply for a situation like that, especially in abusive households. The major reason why shootings are so common in america is a matter of ease. If they aren't going to use a gun, they might use a bomb, or a knife, but those things are either easier to track, or easier to stop, or harder to execute. And that's really what it comes down to. Reduce the problem one step at a time, rather than imagine you'll solve everything at once.
As for other major factors, everyone is 'mentally ill', that's not really an excuse in this day and age. You can easily claim that shit after the fact, so why does it matter. Violent media and bloodsport have existed forever, and restricting that by any degree will create blowback that creates a greater problem than the one you're trying to fix. America could try and fix the symptoms, or just wait till the cancer does it's job and kills the country properly.