I think you covered all the reasonable solutions. Common sense background checks, restrictions on criminals, etc.
The single shot restriction certainly sounds good, but let's not forget they still haven't enacted a federal ban for bump stocks even after Vegas, and I feel like some entrepreneurial mind would find a similar solution for other guns.
So now we can discuss more drastic measures, right?
How about relaxing gun control? Make it cheaper & easier to get guns. It's said in many places (around the world) it's easier to get a gun than it is a cellphone. Imagine if they made guns as ubiquitous as phones, or drivers licences. We know the NRA won't have an issue with it since they flouted the idea of arming every schoolteacher in the country, and they often spout the mantra "The only way to stop a bad guy with a gun is a good guy with a gun". So let's go all out. Make it "normal" for anyone over say, 16 to have a gun. Get rid of the stigma. Pacifists could be the new vegans.
Wouldn't criminals think twice before committing gun crimes if odds were everyone could fight back with their weapons? And even if they didn't, you would still have the average Joe able address the situation on their own.
Cost? Smartphones were only for the rich a couple decades ago, and now preschoolers have them.
Training? Treat it like driving; make it a right of passage for teenagers.
Half sarcastic, half serious. Wouldn't it be a viable solution, at least on several fronts?