Disclaimer: I'm in favor of gun control.

So here's why neither of those solutions are going to help a damn thing.

First, background checks. Background checks are only as good as the funding behind them, and I will bet my bottom dollar that if laws requiring stronger background checks are implemented, the top priority for the GOP will become defunding those programs. That's assuming the GOP doesn't manage to sabotage the program from the start. Government oversight is only as good as the weakest link, and unfortunately the GOP has made it their mission to locate and exploit as many weak links as possible in order to 'prove' their mantra that big government doesn't work.

Second, preventing people with mental health issues from obtaining guns. This is even more impossible. How do you force people to undergo psychological testing? You can't rely on self reporting, that would solve literally nothing. If you live with an immediate family member with psychological issues, are you not allowed to buy a gun? What if you already have guns before you or a family member are diagnosed? Are you required to give them up? If you have a doctor declare you 'cured' or 'misdiagnosed' do you regain the right to own firearms?

Gun violence in America is a symptom of greater issues- massive negligence over mental health, rampant corruption/incompetence in local governments, deepening racial tensions, and a rising percent of the population living under the poverty line, for starters. Here's the reality- mass shootings are horrific, but make up an extremely small percentage of gun deaths. The huge majority of gun deaths are by suicide, and most of the remaining gun violence occurs in low income areas, especially those that are predominately non-white.

We cannot pass inevitably arbitrary restrictions on gun ownership and call it a day. Those restrictions will prove ineffectual sooner or later and then will be promptly rolled back. The root causes of gun violence need to be addressed first and foremost if we expect to accomplish anything of value.

But, there are restrictions that aren't arbitrary. There is no reason for civilians to have access to automatic weapons, period. Same with silencers, armor piercing rounds, and the like. People want these things the same way they want faster cars and bigger houses, and that isn't good enough. If folks want to go to an authorized firing range to play with machine guns, then I'm all for it, so long as the guns are owned by and remain on the range. Almost everyone except for the NRA thinks this is reasonable, and we should start there.