(This is all naturally just my own understanding of an incredibly complex topic.)
Mental illness isn't some entirely distinct entity or disease that overtakes the body like a cancer or parasite. It is a failure to regulate within what we deem 'normal' bounds of neurological and behavioural homeostasis. This failure to regulate is complex and still not well understood. Some 'disorders' are due to deficiencies in specific neurotransmitters, as far as we can tell. Others may even be related to gut biome or general inflammatory response of the body. We know that 'mental illness' in later life is highly correlated with exposure to adverse childhood experiences. We also know that environmental factors such as sunshine, nature and bodily movement somehow affect the prevalence and expression of behavioural disorders. Why should all of this be the case if 'mental illness' were just a simple disease?
"Mental Illness" unfortunately grossly oversimplifies a set of different things. It's like characterising a person with a broken leg, another with cystic fibrosis, and another with a stomach ache due to over-eating all as simply "bodily ill" and ignoring the vast differences in prognosis and treatment between them.
Suicide like all other behaviours humans immerse themselves in is complex and not clearly attributable to any specific cause or set of circumstances. There are correlations that would invite theories but it is not a simple thing. As euthanasia rights increase across the globe, as well, you can see old and young people alike taking a cognitive view of their status and concluding that death would be the best course of action in light of extreme and chronic pain. There is no singular defining distinguishing factor between the cold-hard cognitive rationale of euthanasia and the ostensibly 'infected' mind-state of suicide. There is not single reason why people commit suicide. If we are to say that suicide is due to "mental illness" then we may as well say that it is due to "bodily disfunction" or "misaligned bodily tissues". Closing off the arena of understanding by oversimplifying a vastly complex behaviour is not of service to reducing that behaviour.