This is a big question that focuses on definitions, and given the millennia involved, this answer is much more of a survey, where it's difficult to go into heaps and heaps of details on any particular topic! But, in a nutshell, it all depends on what we define as psychology. Your focus here is on psychology is clearly a bit more on psychotherapy, but psychology is a broader field than therapy, focused on studies of the mind more generally - how do we remember things, for example, or how do people behave in groups?

If you go for the standard definition of psychology that's in most post-1960s textbooks - the 'scientific study of the mind and behaviour', or some variant on that - it's implicit in the definition there that an activity like psychology needs to be conceptualised as a science for it to count as psychology. If so, we're looking at the 19th century at the start of scientific psychology, more or less - William Wundt in Germany and William James in the US both had laboratories in the 1870s dedicated to scientific studies of psychology (though neither was focused in their research on mental illness). This is - not coincidentally - relatively soon after the publication of Darwin's On The Origin Of Species, which implied that human minds were a product of the natural world, and so therefore should be able to be studied by science, which studies the natural world. Earlier modern philosophers, since Descartes, had wrestled with the seeming dualism of a mechanical, 'clockwork' universe, and a human mind/consciousness that seemed somehow non-material, and, from that point of view, Immanuel Kant had argued specifically that a scientific psychology was impossible, as the topic matter of psychology was separate to the topic matter of natural philosophy/science.

Of course, if you read the contents of a modern psychology textbook, the topic matters discussed aren't so different from John Locke's book An Essay Concerning Human Understanding from 1690, which covers much the same ground, except with much less reference to experiments that have been done; Locke is much more inclined to use reason to lead to his conclusions on the topic matter. And Locke, of course, built upon previous philosophers who had things to say about the human mind/soul/psyche (for all that he's associated with the tabula rasa/blank slate theory of how humans acquire knowledge, he didn't invent the metaphor by a long shot); as you say, the stoics certainly had detailed theories about what caused human misery and how best to overcome it.

As to the treatment of mental illness, unsurprisingly ancient people did attempt to treat what we would now consider mental illness; /u/Bentresh here discusses a Hittite text from the Late Bronze Age which is structured in a way involving both diagnosis of a particular set of symptoms and a cure; it's just that, to modern eyes, the treatment recommended is fundamentally spiritual in focus, with an emphasis on ritual and being in the good books of the Gods.

The modern treatment of mental illness from a medical perspective, premised on the assumption of natural rather than supernatural causes, has roots in Ancient Greek/Roman writers like Hippocrates and Galen, which generally saw mental illness as being caused by imbalances of humours (bodily fluids), and thus recommended ways to balance the humours as a way of treating the illness (leading to patients being bled because they were suffering from mental illness). The modern English word 'melancholy' comes from the Greek word for 'black bile', which doctors in the Hippocrates/Galen tradition believed was a humour associated with depression (but also other illnesses - doctors in this tradition believed that depression was usually associated with other illnesses and didn't really treat it separately). This way of thinking about mental illness from a medical perspective was deeply influential for millennia.

It was really only in the 19th century, with the rise of the 'alienist' - the specialist doctor working in mental health, often in asylums, that you start to see a move away from this kind of humour-focused treatment of mental health and towards treatments of mental health that look more like they do today, with emphasis on medicine and talking therapies. In the late 19th century, post-Wundt and James, you see the transformation of the alienists into psychiatrists; Emil Kraepelin, for example, who is famous for the diagnosis of dementia praecox (e.g., schizophrenia), was a student of Wundt's as well as being a medical doctor; the likes of Kraepelin and Sigmund Freud bring a new scientific rigour (of sorts, from today's perspective) to the medical study of mental health. The field of academic scientific psychology started by Wundt and James only really started to focus on the treatment of mental health after World War II. At this point they were asked by authorities to use their psychological skills to develop treatments for veterans of the war suffering from trauma - the scale of the war and thus the trauma being dealt with in the aftermath having swamped the ability of psychiatrists of the era to respond. The field of 'clinical psychology' was effectively developed in response to this need in World War II.