So then you have to go around saying that serial killers are the most healthy people there can be and 100% rational. Surely this is a mistake?

Of course we have mental illnesses that describe things like this, like "antisocial personality disorder". If anything, all we need to do is invent "wants to shoot people personality disorder".

Perhaps the wrongness is that the murderer is simply evil. I don't have a framework for helping evil people nor positively identifying them before they commit evil acts. If someone does evil, they can be punished to dissuade further evil acts, caged to protect others from them, or killed. It is an injustice to do any of those things to a person before they commit an evil act.

If, however the wrongness is an illness, it might be possible to diagnose them before they do great harm to others and provide treatment. Of course there's a dangerous slippery slope possible here in which anyone who's at all atypical is given involuntary "corrective" treatment.

Media reports often assume a binary distinction between mild and severe mental illness, and connect the latter form to unpredictability and lack of self-control. However, this distinction, too, is called into question by mental health research. To be sure, a number of the most common psychiatric diagnoses, including depressive, anxiety, and attention-deficit disorders, have no correlation with violence whatsoever.18 Community studies find that serious mental illness without substance abuse is also “statistically unrelated” to community violence.40 At the aggregate level, the vast majority of people diagnosed with psychiatric disorders do not commit violent acts—only about 4% of violence in the United States can be attributed to people diagnosed with mental illness.41,42

The media's position is untenable. If your definition of mental illness doesn't include adolescents who want to murder their peers, then it's defective by design.

Your first link doesn't address mass shooters.

The second one (written by an "enterprise reporter" and quoting from forensic psychiatrists) draws an astonishing distinction between "mental illness" and "sociopathy or antisocial disorders" (!), and defines most mass shooters as sociopathic and therefore not mentally ill. I don't know how anyone could read that article and be convinced by it. I should note that forensic psychiatrists are by far the most conservative (not in the political sense) practiciners in their field, a shocking number of them are quacks, they often have little experience treating patients, and they tend to believe in more black-and-white/"predictive" (and excessively confident) theories, suitable to their role in courtrooms.

Under the old-school DSM mindset, if you're not diagnosed with anything, you are defined as healthy. That type of thinking will never allow us to truly address society's emotional health. Emotional health is a spectrum; we're all at varying degrees of emotional health, and the average (i.e. "normal") person is significantly emotionally unhealthy.

Any reform that aims to genuinely improve society's emotional health has to move beyond the old model; and in any model of emotional health (where low self-esteem, social anxiety, low emotional self-awareness, attachment disorders become part of the picture), any shooter would be defined as severely emotionally ill; moreover, any reform that aims to detect attachment issues, low self-esteem and more, early on in life, in a systematic way, would prevent most (if not all) of these shootings from happening.