But the Right's case isn't so great, either. Yes, we should have better mental health policies. And it's pretty clear that the Navy Yard shooting, at least, could have been prevented with more attention to mental health. Evidently the shooter had displayed clear signs of mental illness when speaking to police just days before the shooting. But the police hadn't called mental health officials. Had the done so it's possible the shooting could have been prevented.
In general, though, would better mental health policy have much effect on gun violence? Not really. The problem is that most people with mental illness aren't extraordinarily violent, and most violence isn't perpetrated by the mentally ill. So not only is it difficult - nearly impossible - to determine who is likely to commit gun violence, even if we could keep guns out of their hands the effect would be only marginal.
This is a problem with no easy, obvious fix.