The only shootings where mental illness plays a major factor are suicides. When it comes to gun violence, only 4-5% of perpetrators have a severe mental illness. When it comes to school mass shootings specifically [ source ]:

• 67% are white
• 100% are male (95% according to a different source)
• “Severe mental illness (e.g., psychosis) was absent in the majority of perpetrators; when present, psychotic symptoms are more associated with mass murders in academic settings involving means other than firearms”

And with regard to school shootings generally:

• 77% of the time, someone knew about their plans for the shooting ahead of time
• more than half of K-12 shooters have a history of psychological problems, but the bigger issue is that nearly three quarters of the time, they had been being bullied or harassed in school
• depending on the source, nearly half or more than half got the gun from home or a relative, often by stealing an unsecured or under-secured firearm
• 91% of shootings were with a handgun

If we could reduce bullying and do a better job at making students feel like they have value and matter, that would go a lot further toward reducing school shootings than anything involving mental illness (aside from, perhaps, efforts to reduce the stigma associated with it).

Substance abuse - drugs, particularly those that are illegal, and alcohol - as well as poverty and inequality is much more strongly linked to gun violence.

I’m not saying that we shouldn’t continue improving our available mental health resources (the majority of deaths from guns are by suicide, after all), but we shouldn’t use mental illness as a scapegoat.