I feel like if America has a mental illness problem, surely it would be addressed by providing more mental health services perhaps cough universally to the populace somehow. But, it's all rhetoric. If claims there was a mental illness problem were in good faith, we'd see American action on mental health.
"Mental illness" is just a convenient way to dismiss any notion of doing anything about gun violence. It's misdirection. Because, as you point out, we're not going to actually do anything about mental illness. Indeed, half the Congress and all of the Administration is trying to reduce access to health care, which would include mental health care.