I want to start by saying that I fully support the banning of assault-style weapons and especially high capacity magazines. This just seems like common sense. But I don't think this gets at the heart of gun violence in America, rather these measures are sort of cherry-picked issues that aren't very controversial among the sane crowd.
The large majority of gun violence in America is done with hand guns. Usually it is poor young men shooting other poor young men. I imagine that most of these guys feel trapped and powerless, unable to make their lives better and so they gravitate to violent methods of confronting the world around them as a way to save their self-esteem. Violence begets violence, and having a gun becomes necessary for survival. Because young angry men (often minorities) are not a group that people feel a lot of sympathy for they will not be the target beneficiaries of federal legislation.
So what to do about it? A lot of things, really, but I suspect the thing that would help the most is doing something about income inequality, providing real health care to the poor, and developing an educational infrastructure that promotes class mobility. Free preschool and parenting classes should be available to everyone. Yes, we definitely need criminal background checks and so on, but wouldn't be even better to do something about the cultural institutions that lead Americans to violence so much more often than citizens of rich countries?