But the issue is complicated by cases of guns accidentally going off
Modern firearms don't accidentally go off. They go off when the trigger is pulled. That is caused by negligent or unsafe gun handling procedures (a la the pilot whose gun discharged.
stray bullets (e.g. through the wall into the kid's bedroom),
What complication? Yes, overpenetration can be a problem, however if the gun is actually being used for self defense, this is usually not a problem unless the person firing the gun is not on target. Don't confuse this with drive by shooting and other illegal firearm uses.
and those less responsible such as kids finding the guns.
You don't take away people's right because someone can be stupid. The same can be said of cars, kitchen knives, powertools and even the keys to the front door. Heck, going down the stairs is potentially dangerous for an unsupervised child.
I can't imagine that many people who would want to own guns would voluntarily receive large amounts of gun training.
You'd be amazed. There is large training class industry. Courses near me book up weeks in advance. People travel to out of state training sites. There are lots of people, who, given the chance, want to learn to use their firearms appropriately. Not to mention the growing popularity of IDPA, IDPA like, IPSEC and other shooting leagues.
I know of someone in a large urban area who recently had a gun from their arsenal accidentally
See accidentally above. If it went off when he pulled the trigger, it wasn't an accident, it was negligent.
(the arsenal was illegal for him, although he could have still legally owned a gun in the city).
So he was willing the break the law? I don't want him living near me either, since he is a criminal.
There's no way I want more people like that living in the four apartments next to me, in a world of permit-less gun ownership in urban areas.
1) Why distinguish urban from rural. Frankly, the need for a self defence gun seems more likely in a higher population density, urban area.
2) The guns are already there, You just don't know it. Maybe with more training (you can't train with your illegal gun) he would have had better gun handling skills.
I know that banning guns outright isn't perfect either, but every policy is imperfect even when they can still be effective. To me, the question is an empirical one of effectiveness, better suited for the criminologists than the constitutional lawyers, as to which degree of right-to-bear-arms in particular areas will give us the least amount of gun-related crime.
Just remember that when you criminalize gun ownership, the only people who care are the ones who care about having a criminal record. You don't disarm criminals. More importantly, especially in the case of Chicago, you don't do squat when the penalties for illegally carrying a gun, even by a felon are essentially extension of probation. Maybe the key is better/more rigorous enforcement of existing "no criminals with guns" statutes then it is to disarm the law abiding.