Other countries that have lots of guns still have fairly strict regulations about them, as far as I'm aware though. I think the mental health argument is just something used by gun nuts to try and deflect blame. It's not as if plenty of people in the UK don't have mental health problems.
But if you're looking for factors beyond just the guns themselves, then it'd be worth looking at the fact that the US has the level of wealth inequality of a developing nation. That is typically correlated with a whole host of social problems, including levels of violence. Add to that their horrendous prison system that basically removes any opportunity to become a success legally once you've been convicted of one crime, driving people into a life of crime (massive generalization, I know), which in the States, is going to involve guns by definition.