Lucy doesn't deny the reality of "suicidal despair, hearing hostile voices, crippling anxiety and mood swings". But she does not think these experiences should best be understood as mental illnesses/disorders. In fact she goes further and says that there's never been any evidence to support the idea that these "very real experiences" are best explained as medical disorders.
What she seems to mean by 'medical disorder' is a condition caused by physical pathology. Here, in a way, she follows Thomas Szasz, who thought mental illness is a myth. For Szasz, it was a misnomer to call mental illness an 'illness', because it's not an illness with physical lesions (see eg. previous post).
I agree with Lucy (and Szasz) that functional 'mental illnesses' are not "chemical imbalances, genetic flaws or other bodily malfunctions". Where I have difficulty with her (and Szasz's) view is that I think the best way to understand medicine is that it covers the treatment of both physical and mental illnesses (see eg. another previous post).
In fact, our modern idea of physical pathology only really developed from the mid-nineteenth century with the correlation of pathological, including histological, findings with clinical symptoms (see eg. previous post). The problem for psychiatry was that it did not completely fit with this development of anatomoclinical thinking, which was of such importance for the advancement of modern medicine in general (see eg. another previous post). Modern psychiatry's always hankered after this identification with the rest of medicine. Most psychiatric illness is functional and not organic, in the sense of not having an identifiable physical lesion in the brain (see eg. yet another post). Lucy's correct that much of modern psychiatry assumes that physical pathology will be found to underly what is called 'mental illness'. It's always tended to believe this, but, in my view, thereby avoids philosophical issues about the mind-body problem and the nature of life in general (see eg. previous post).
I also agree with Lucy that psychiatric assessment is about formulation. It's not commonly appreciated that the standardising of procedures for history taking and mental state examination in psychiatry at the beginning of the 20th century was about formulation (see eg. previous post). What should be important in psychiatric assessment is helping people understand their problems. Reducing people's problems to brain disease has the potential to be stigmatising by turning people into objects.
I, therefore, understand Lucy's concern about psychiatric classification. I agree with her critique, if the aim of psychiatric classification is to create a functional diagnostic system based on biological markers. This was the original intention of DSM-5 and it failed (see eg. previous post). But Lucy seems to think that psychiatry needs to have a reliable and valid classificatory system to be a science. Following Peter Breggin, she argues that psychiatry is undermined by not being able to treat mental illness as physical illness.
Here, again, I have difficulty with her view. Any psychiatric classificatory system is merely descriptive, not aetiological, certainly not biologically-based in the sense of brain abnormalities (see eg. previous post). It needs to be understood for what it is, and there may well be nothing to be gained, or even harm caused, by using psychiatric labels. But sometimes it can be helpful to use words to describe mental states. This was in fact how modern psychiatry first started from the end of the eighteenth century by attempting to reason about madness (see eg. previous post). There will be inevitable uncertainty about using these descriptions of the ways in which people react to the situation in which they find themselves (see eg. another previous post). Of course brain abnormalities can cause psychotic symptoms as part of delirium or dementia (see eg. previous post). However, for the vast majority of psychiatric presentations, which are functional, we have to rely on our ability to identify patterns of responses if we are going to create any validity for a psychiatric diagnostic system (see eg. previous post). We are profoundly limited in what we can achieve and too much should not be made of these limitations.
I just worry that Lucy is making too much of this situation. Mental health practice is interdisciplinary but that doesn't mean it's not medical. I just think it's potentially misleading to suggest that what's identified as 'mental illness' may not be a medical problem. Medicine includes both physical and mental aspects.