Many mass shooters are/were either severely mentally ill (many were extremely psychotic, depressed or some mixture of both) or have severe personality disorders (narcissism to full-blown psychopathy).
The Columbine killers are an example of this. Harris was a full-blown psychopath full of charisma and charm who lied to everyone while producing some of the most narcissistic, hateful screeds you can imagine in his private journals where he gloats about lying to everyone and believing he has the right to do what he wants to anyone, including kill most of the human race. Klebold was a suicidal depressive who planned to kill himself, anyway. He appeared to be a "sensitive sort" who fell apart over time. (All the weapons used in that attack were illegally aquired and/or modified, too. Not to mention the homemade bombs and other explosives that they intended to kill hundreds with.)
Peter Langman is an expert on school shootings, but his work goes beyond just them. His psychological profiles of them (he's a psychologist) are very illuminating:
School Shooters .info | Resources on school shootings, perpetrators, and prevention
But, no - being a murderer doesn't mean that you are necessarily mentally ill or that mentally ill people are a disproportionate danger to others.
Another factor to consider, is that many of these shootings are actually a form of grandiose suicides. They wish to vent their rage at the world. "I'm going to die and take as many with me as I can!" is that mentality. Multiple of these shooters were failed by society and fell through the cracks as their mental state demonstrably deteriorated over a number of years (such as Lanza, Cho, Holmes and others). Others had really awful life histories full of severe trauma, such as teenage Jeff Weise, and they get sucked into a gradually intensifying abyss of despair, anger and hatred. Those types could've and should've been helped as there were many warning signs that things were going very wrong with them.
(This is also very much a male thing. Female mass shooters are almost unheard of. It tends to require a certain fascination with violence and destruction that females aren't generally known to have.)
So mental health does need to be part of the discussion, but in a more nuanced and informative way, not in a "crazies with guns" way. This is really a symptom of social breakdown. Something is wrong with this society that it is producing such violent, nihilistic individuals, and in such numbers. (There's many who feel the same way but either never commit the act or are caught before they can.)