Consider addiction to smoking. There are plenty of people who picked up smoking for no other reason than 'fitting in' or 'curiosity'. Which I'd hardly call a mental illness. Many of these people kept smoking, and sadly probably many of them died from smoking.
Similarly, I suspect there are many alcoholics who slowly became so because they entered a culture of alcohol.
Addiction is a powerful thing in itself, and while the solution to many might be uncovering the underlying mental illness, for many it might 'just' be a matter of unlearning a powerful, destructive habit.
Of course in both cases professional help is necessary, but treating an 'underlying mental illness' in cases where there are none can also have bad consequences.
Among the many addicts and ex-addicts I've known, there have been more than a few that were actively treating a 'deeper underlying mental illness' that I suspected didn't really exist in the first place. And while that's probably better than being an addict, it's still not good.
in fact, this is one reason I have issues with the approach Alcoholics Anonymous. They seem to treat all alcoholics as sufferers of a lifelong disease, and I cannot help but wonder if being constantly reminded of a thing you don't want to do anymore is a good solution for all people. For some, sure. Some people might have the 'alcoholism gene'. But I don't think that's always the case.