We know that AD has a specific course, with specific and non-specific signs and symptoms. It has demonstrated changes in the brain at a microscopic cellular level as well as a macroscopic level when it is advanced. The gross brain of a victim with advanced stages of AD looks different than the gross brain of a person without the disease.
In many psychiatric disorders you can not find the physical changes in the brain. That does not mean they do not exist, it is just that science has not caught up in certain psychiatric disorders.
For a long time when AD was first becoming known in our conscious awareness some twenty or so years ago, the issue of a psychiatric disorder or not was a hotly contested and emotional issue by many advocates and groups. It really stemmed from the fact that we have terrible biases and stigmas associated with mental illness. In fact we still do. We did not want to classify our AD victims with 'those' kind of disorders.
In fact there are many psychiatric disorders where you can find differences in the brain. It is well-established that people suffering with schizophrenia for example, have larger ventricles in the brain compared to people without schizophrenia.
The debate was further compounded by the fact that the Diagnostic and Statistical Manual (DSM) which is still on its 4th edition and goes back to the early 1950's in the first edition, lists Alzheimer's Disease. This manual is sort of the "Bible" of psychiatry developed by committees of experts that have time, and published by the American Psychiatric Association. It lists diagnosis and criteria to make the diagnosis. It has two actual practical purposes besides academic (and political). lawyers and judges love it, in a court of law, because it is in black and white and considered dogmatic, and insurance companies like it because they recognize some diagnosis of which they will actually reimburse for.
Many people do not like that Alzheimer's disease is listed in the DSM.
The fact of the matter is there are many personality changes and behavioral changes in AD. A victim can develop depression or display irritability, aggression, or speak in a delusional fashion (for example become paranoid) and even develop hallucinations- seeing or hearing things that are not real.
These are all signs and symptoms we generally reserve as conventional psychiatric symptoms.
Sadly we really don't treat our mentally ill with much respect or dignity in our 21st century world. The stigma abounds. It is not unlike how society stigmatizes AD, whether we want to admit it or not.
As families we often don't like it when the psychiatrist gets involved in the treatment of our loved one with AD. The psychiatrist is usually associated with the prejudices we have about mental illness.
In the end the question is irrelevant. Mental illness is real and people suffer with it, just like AD. both need to be treated and they both need to be treated with more compassion, no matter what way you look at it.