Background checks do nothing if somebody has no criminal record. Their are loads of murderers, rapists and pedophiles that could pass a background check with flying colors since they haven't been arrested yet.
As for psychiatric checks someone could just lie. It isn't that hard to do a bit of research on what the doctor wants to hear. A psychologist can't read minds after all. They are not nearly as infallible as people like to think. Someone can be perfectly sane and still be a terrible person.
Exactly. Just to cite one example from up North, the colonel who was the Canadian equivalent of the command pilot of Air Force One turned out to be a serial killer. Nobody's screening process is infallible.