Technically by narrative description in the DVM manual a sociopath doesn't really have a high opinion of their life to begin with so public hangings aren't going to somehow drop gun violence.  

If the death penalty worked Texas would have no murderers. In fact Texas has a fast line to the gas chamber and still results in a long line of murderers.  

Death doesn't discourage people from killing.  Especially not when murders are majorly crimes of passion.  Get someone's emotions going, hand them a gun and see what happens.  

Even the most law abiding have fallen and people said but "he/she seemed fine". They had no real problems blah blah blah 
Emotional reactions + weapons are usually never positive.  
While his moral caliber might be what you're putting on the line here, it's a bit late to think you're going to rewind the clock to a somewhat "better" moral time.  

I think folks had better start coming up with new ways to solve their problems.