"This is not a "gun" issue. This is a legal system and mental health system issue."
Respectfully, it is both. The kids were killed by a madman with guns. We need to try to prevent madmen from getting guns. It seems to me that one positive step - which has nothing to do with banning anything - would be to make sure law-abiding gun owners are not unknowingly selling their guns to people who are "mentally defective."
No, a background check would not have prevented this particular crime. It might, however, prevent a future crime. One more massacre like Sandy Hook and you can be sure that our gun laws will look like Great Britain's in short order.