Mental illness account for a very small proportion of mass shootings in the US

Honestly I would argue we need to expand our definition of what a mental illness is, in order to help these folks get treatment before they go and shoot a bunch of people.

With that being said I think it's important to call out this distinction for two major reasons. For one, it shines a light on the fact that someone having any mental illness does not mean they are going to go on a mass shooting. For two, it also shines a light on the fact that the people who do break in this way aren't understood well - if they don't classify neatly into the existing mental illnesses we have, we by definition do not have good research on the subject. Do patterns exist among this population of mass shooters that we can identify? Are there specific ways that they think which are maladjusted and can be corrected through existing education and mental health offerings? Might they belong to a larger group of illness which also includes other folks who are suffering in other ways, where studying this group might shed light on other folks who need help?

Don't worry, you can mass murder with a knife too.

They call it out a little further down.

Half of all mass shootings are associated with no red flags—no diagnosed mental illness, no substance use, no history of criminality, nothing. They’re generally committed by middle-aged men who are responding to a severe and acute stressor, so they're not planned, which makes them very difficult to prevent.

So they are not necessarily in a good emotional state, but they do not have a mental illness.

While I think this is a reasonable sort of surface-level interpretation, I think it misses a bit of what typifies mental illness versus just being destructive, malicious, desperate, or extremely entitled.

Mental illness is something your brain is doing to you. It's not just a thought that you have and roll with, it's a persistent pattern that you struggle against. Just deciding that the thoughts and feelings being produced are inaccurate or unhelpful doesn't make it go away. It's not just extreme emotion, it's emotion that's being switched on in a way that isn't tied into the continuity of your more volitional patterns of thought and feeling. It's not just that the thoughts and behaviors playing out are unhealthy.

To put it into metaphor, think of your life and your interactions with the world like a video game, with your brain being essentially your character controller, interpreting your actions and bringing them into the world. You can decide to do healthy or unhealthy things with your character, but those things are under your own volition. Mental illness, then, is like a poorly coded character controller throwing errors and causing unforeseen bugs. Like, for example, if I push the down button there's a 30% chance that I randomly move to the left first, rather than moving in the appropriate direction.

That 30% chance might send me careening into a pit, but chances are that once I'm used to having this bug, I'll be aware enough of it to try to compensate. It might not always work, and I might drift a little left occasionally, but if I give myself a bit wider berth for any obstacles on my left, I'll probably be okay. This is distinct from someone who uses their volition to throw themselves into a pit on purpose.

Are both potentially bad for the character's health? Yes. But only one is caused by a character controller error, and because my goal isn't 'throw myself into pit', I'll probably do a much better job avoiding pits than someone who's jumping into them intentionally. These two problems are fundamentally different in that one is a product of a person's volition, while the other is a problem with the means by which they interact with the rest of the world.

That's not to say that people with mental illness are going to accidentally assassinate someone because they pressed down and went a bit left, but it illustrates the fundamental difference in making a bad decision versus struggling with errors in your brain.

That someone jumps into a pit on purpose does not imply that their character controller is bugged, especially if they smoothly beeline it while showing all signs of acting with intention.

I was never under the impression that most mass shooters do it for fame or notoriety, but rather as revenge for some perceived injustice from society at large or by the particular community they attack.

Even still, it's very far from a healthy mindset.

One of the key features of scientific studies is that they only go as far as the data and the experimental design can support. Because of this it is not unusual for a study to give results that may be surprising or counterintuitive, but not be able to explain why. This is especially true for complex issues. I'm not trying to dismiss your question, and I think several good ideas and observations have been made in the ensuing comments, but in the end I don't think this study which describes the correlation between two factors, can give you the answer you are looking for.