For Alzheimer's you state:
"It is in the interest of an individual not just to produce offspring, but also to aid the offspring in producing more offspring as much as they can. So Alzheimer's can still be considered to have the affect of reducing the probabilty of having your genes present in future generations."
If this is your standard, then yes, there are mental diseases. Suicidal depression, for example. Paranoid schizophrenia. Heck, severe untreated obsessive-compulsive disorder is going to impede all sorts of things, and these can often (or even preferentially) strike during prime reproductive years.
While getting one "necessary and sufficient" definition of disease is going to be difficult, I think, any definition that excludes suicidal depression, paranoid schizophrenia, and obsessive-compulsive disorder is severely lacking.
Now, I am utterly sympathetic with the narrower point that many named "disorders" are probably undeserving of the status, especially many "personality disorders". There's a couple of reasons. One is that the field is still developing. There's great progress in understanding certain subsets of mental problems, but a murky mess at best in many more.
Another is the realities of health care in this country. In the U.S., unless you have a diagnosis, it can be hard to get reimbursed for treatment. So, the whole DSM system for diagnosing mental disorders is in part driven by the need to have some sort, any sort, of diagnosis available for someone who is receiving treatment.
Many people probably do need psychological help, and will benefit from it, even though they don't have a specific "disorder" or "disease". But try to get a health insurance company to keep paying for therapy for general coping skills, or whatever. Not gonna happen in most cases.