Define “mentally ill”.
That right there is a problem, especially as many things that qualify as “mental illness” are not things that lead to shooting rampages. Depression is a mental illness, but one that would probably deprive one of the motivation to even pick up a gun, and a higher likelihood of turning it on themselves without harming others. Some would consider migraines to be a mental illness.
Plus it opens the gates for abuse; for example, finding credible scientific evidence of the link between sociopathy and voting Republican and using that to declare Conservatives “mentally ill”. (~Not that anyone would ever abuse power like that.) It would require some pretty massive oversight to keep it from getting tyrannical.
Personally, I feel that “mentally ill” is somewhat context sensitive. In the context of killing people, I would define it broadly as, “an imminent physical threat to the safety of others for no just reason”. If one is under the delusion that people with glasses are alien invaders from Betelgeuse and wishes to use their second amendment rights to save humanity by plugging us four-eyed people, then I think it safe to say that that person is mentally ill. If, on the other hand, you have an otherwise upstanding citizen who wishes to carry a sidearm for protection due to a distrust of the effectiveness of police, that person is likely merely a concerned (but mentally healthy) citizen.
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”[D]o non-mentally-ill (or perhaps, rational) people ever kill?”
Fairly often, actually. That’s pretty much the definition of Voluntary Manslaughter, a.k.a. Third-degree Murder.
”[D]oes killing people (especially a mass killing) by definition make you mentally ill?”
Not by definition, though it often does lead there. Taking a life is something that leaves a mark on you, and some people can’t handle that strain even in the most justified of killings like self defense. It’s rare to find someone who has killed another human under any conditions and not been scarred by it. Some feel intense guilt that leads to depression. Many police and military vets wind up that way. Others try to save their sanity by becoming detached and less empathetic (some would argue less “human”).
Still, there are others who can kill while still having enough respect for the sanctity of human life to not be a sociopath.
”[I]s labelling someone ‘mentally ill’ in this context masking a deeper societal problem?”
Very much so, and it’s something that we’ve barely seen the tip of the iceberg on.