It's wrong to suggest that Americans don't want to change the law regarding common sense gun legislation, the majority of Americans were supportive of Obama's background checks and restrictions, (the first steps after Sandy Hook) most people are fine with common sense legislation and most are fine with a ban on assault style rifles, it simply is not the case that people are against any form of legislation - maybe out in the sticks this is true, but overall it isn't.
The problem is that the government are working for the NRA and not the people, so nothing that the people want gets enacted.
Common sense legislation that would go a long way to solving this problem would go along the lines of;
Firearms licensing and age restrictions (minimum 21 year old) which isn't much different from alcohol purchasing..
Social media and mental health screening for license applicants
Full background checks on all purchases
Revoke the Brady law (gun show loophole) with strict penalties for law breakers
Outright ban on semi-automatic assault style rifles and bump stocks
Consider that out of the above elements, some of them would be enough to enable the TSA and Interpol to put someone on a no-fly list, but nobody would bat an eyelid if that same person were to buy an AR15..
Another big thing that I'm convinced would help, would be a media blackout on mass shootings - I understand it's practically impossible, but I was here (LA) during the Vegas massacre, the church shooting and now last weeks shooting. The news coverage has been psychotic, it gets turned into a week long event - we see everything, the killer, who he was, what he wrote on facebook, videos, pictures, background, court appearances, the media lead with the body count - it's just adding to the climate and it makes further shootings more likely, it's very bad indeed.