I'm not sure suicides strictly are the result of mental illness. I, for example, have an uncle that killed himself. He may have been clinically depressed, but mainly, he was an alcoholic. I think it's likely what depression he had would have resolved if he successfully dealt with his substance abuse problem.

Did he die because he was mentally ill, or because drinking can be a pretty bad problem?

I guess it depends on whether you consider alcoholism to be a mental illness. It is listed as one in the DSM-V.

I'm drawing a distinction between "mental illness" and "mental illness diagnosis" that I maybe didn't make entirely clear. I think the former is increasing, but the latter is increasing faster than the former.

It seems like only around 10% of suicides are not associated with mental illness. So I agree it seems implausible that this 10% would account for the entire increase, but it could be some of it.