As for the opposite, no, if the child is unstable then knowing they can gain access to guns could only make them worse, as you pointed out before, it is the parents responsibility to know if their child is a danger or not, and this is quite clearly not always the case.
How can you know if someone is unstable or not? Some of them hide it pretty well... that is the big main question. Since we stop placing people in mental homes or getting help for people who have mental problems, things have spiraled out of control. I sometimes think that the government actually planned for this when they started shutting down mental institutions and stop giving help for people who is mentally unstable. I am not a conspiracy nut, but it is starting to feel like this. The government and politicians are capitalizing on this big time, pushing for gun control harder than they ever pushed for before.