Except that you WON'T satisfy people who think that background checks work, specifically becase they DON'T work.
I've said it repeatedly on this forum and I'll say it again. Our government is DELIBERATELY not doing it's job, which is to keep KNOWN violent people removed from from free society. By instituting a background check system (which by definition only works to flag those people the government ALREADY KNOWS are a danger to society), the government is telling it's people FLAT OUT that it is unwilling to do it's job. The people of this country need to understand this, and to demand the government REMOVE THESE VIOLENT PEOPLE FROM SOCIETY, UNLESS AND UNTIL THEY ARE NO LONGER A THREAT TO FREE PEOPLE!!!
It's so simple, and yet very few people understand that we aren't even having the right debate. It should be about STOPPING VIOLENT CRIME, and not about guns at all.
Now with regard to stopping people who suddenly "snap' but have no prior record of violence, there truly is no way to prevent this from happening in the first place. It is simply a risk that is inherent in a liberty-loving free society. That said, the damage caused by such individuals can be minimized when law-abiding citizens have the means to shoot back and aren't legally restrained from doing so.