I personally think the issue is more nuanced than this. It’s not that “no mass shooters are mentally ill” — many of them are. Many of them do have some kind of issue, whether it’s depression, bipolar disorder, schizophrenia, psychopathy etc. I watched a documentary on Eliot Rodger where it went into detail with his mental health struggles and his lifelong depression. This isn’t a conspiracy.
I also have personal experience with mental illness (albeit complicated by drug use) that led to violence. One of my loved ones fatally shot someone while hallucinating and I was a witness to his mental state that night. So yes, mental illness can 100% be a factor in violent crime.
HOWEVER it is equally true that many mentally healthy people are violent, and that most mentally ill people are not violent. It’s also true that many other countries have similar rates of mental illness with many fewer shootings.
There are therefore many other factors at play that lead people to commit violent crime such as mass shootings, and while mental illness may be a factor for some of them, so too are many other factors such as social alienation, toxic communities such as 4chan and incel forums, racism, misogyny, entitlement and access to weapons with which to carry out their violent intentions.
I actually think that for my loved one, not having been allowed to have access to a gun would have meant he wouldn’t have been able to use it to shoot anyone that horrible night when he was out of his mind. So I do support strict gun control for people with certain kinds of mental illnesses.