Mental illness is not detectable by an untrained bureaucrat. A lot of these people look complete normal on the surface. And the most evil people in the world - psychopaths - are just evil - they're perfectly sane by any psychological test you could care to administer to them. This is just feel-good nonsense that doesn't address the fact the cause of the VA Tech shootings wasn't due to the killer being able to acquire guns but rather that the VICTIMS couldn't.
"Show me just what Mohammed brought that was new, and there you will find things only evil and inhuman, such as his command to spread by the sword the faith he preached." - Manuel II Palelologus
Mental illness is not detectable by an untrained bureaucrat. The VT shooter, on the other hand, was correctly diagnosed on more than one occasion as being a risk to society. It does seem reasonable to expect judgements from appropriate medical personnel to be available to the screening process.