The dangerous thing is to presume that's not true. Or even worse: to lie about it. The phrase "chemical imbalance" comes to mind. Complete and utter fabrication of a physical cause.
It may be that a physical cause is discovered for some conditions which today are considered "mental illnesses"... say, schizophrenia. Well, when that happens it will stop being known as a mental illness and just a physical illness.
No, it's not, it's a subset of physical illness. It's not even possible for it to be otherwise without magic.
Perhaps consider: what things would someone who calls some things “mental illnesses” and not call them “physical illnesses”, contrasted with the things they would call “physical illnesses” and would not call “mental illnesses”. What name would you give to these two categories?
I am reminded of a conversation I had with my dad, perhaps 5 years ago, where he referred to something as a plant, and I was confused because he meant that in a way that excluded trees, while I hadn’t interpreted it that way. Now, of course, in the technical sense, trees are plants, being members of the kingdom Plantae. But at the same time, if one is in an area with 5 trees, and a plant which is not a tree, and someone tells you “look at the plant”, the pragmatic way to interpret what they mean, is that they want you to look at the plant (i.e. the plant which is not a tree).
Scott Alexander said (as a paraphrase/snowclone of Christ’s “The Sabbath was made for man, not man for the Sabbath.”) “The categories were made for man, not man for the categories.”.
Edit: small formatting fix
I think GP's point is that you can't determine a mental illness with physical characteristics/measurements. Can you diagnose schizophrenia with a blood test? X-Rays? Nope. Perhaps an MRI? Maybe, though AFAIK this still isn't really possible.
With physical illnesses it's easy. Broken bone. Cancer. Viral infection. All of these can be diagnosed without communicating to the patient at all, simply by making physical observations using chemistry and physics.
Not so with mental illness. With mental illness you basically have to just talk to the patient, take their word at face value, and then deduce based on symptom pattern matching. A determined fraudster could fake a mental illness, and it would be impossible for doctors to be able to tell AFAIK (correct me if I'm wrong), the same which cannot be said for tumors.
Then it was removed. What happened? Did it stop existing? No. Were homosexual individuals "cured"? No.
Quite simply, the psychiatric community decided that individuals with same-sex desires were not, _and never had been_ mentally ill. The illness was no more.
The same is not true of physical illness.
Different argument:
The diagnostic criteria for "mental illness" overlap. So it's possible to be diagnosed with say, ADD and "Bipolar type 2" and "generalized anxiety disorder".
Which is it? Which illness does the patient have? How does one decide? Which _physical_ mechanism is responsible for one vs the other two?
Physical illness has no such ambiguity.
Nobody knows. Because the map does not map to reality. It's like asking if a hot dog is a kind of sandwich. Maybe. Who cares. It's the wrong question.
Last example:
People suffering from grief/bereavement match many, if not all, the diagnostic criteria for "Depression". Are they depressed? Or are they merely grieving? The answer (if you're a psychiatrist) depends on which volume of the DSM you use (the latest one does not exclude bereavement from the diagnostic criteria).
Is there similar subjectivity with physical illness? No. So they are clearly categorically different kinds of diagnoses, and Thomas Szasz's argument is that for that reason we should not use the word "illness" to talk about these kinds of very real problems.
Autopsies of mentally ill individuals (no history of drug use) and chronic drug users that developed mental illnesses have found many of the same structural defects in the brain.