I am studying to be a clinical psychologist, which is in the same wheelhouse as psychiatry. A lot of people in my academic cohort have had mental health issues in the past, or have lived with them in their families, it's true. However, the people who have been through mental health issues are going to be, imho, the best psychologists. Just from my own observation, they are extremely kind and empathetic, non-judgemental, and feel really passionately about helping their clients. Provided that they're properly educated about boundaries during their supervised practice, and maintain enough insight to know which illnesses are okay for them to treat and which they should refer because they'll be too triggering, I think they'll do great.

There's also a small part of my cohort who have had very little exposure to mental illness/really aren't that interested in mental illness, and are obviously studying psych because they like the idea of being able to "psych [people] out!", aka manipulate them. I'm pretty frightened at the idea of these people going into clinical psych and kind of hope they decide to go into occupational psych or something instead. I definitely think that they're not likely to be better psychologists because they're "sane", ie. they have no history of mental illness. In fact, some of these students have had the most trouble letting go of their stigma/preconceived notions about mental illness in class discussions. For the ones who continue into clinical psych, I can only hope that their supervised practice either straightens them out or turfs them out.

I would guess that people like that are the significant minority who do stuff like violate sexual boundaries with clients. As for mental health issues on the job - while many people going into psychology/psychiatry already have a history of that (as mentioned), I think that depression and other mental health problems can be brought on by the highly stressful job that psychologists/psychiatrists do as well. It's very hard to have 15-20 people per day walk into your office and vent all of their problems to you, and empathize with those people and try to help them. Mental health professionals are trained to cultivate professional distance and to leave their work at work, but they're still human - they still care about their clients and can't help but empathize with them and feel for what they're going through. Even the best psychologists/psychiatrists sometimes have trouble coping with it all. And that's not even bringing into the picture the fact that inevitably a few of a mental health worker's clients are going to attempt or complete suicide - studies have shown that the amount of grief and trauma a therapist goes through after losing a client is similar to what people go through after losing a parent or loved one.

I guess tl;dr, a history of mental illness does not necessarily make a psychologist unfit for their work (it may even make them better at their work), and - more importantly - the stress of the work can also cause mental health issues, even in a person who has no history of them. It makes sense that psychs are overrepresented when it comes to mental health issues, compared to a sample of all other doctors - which includes, say....allergists or orthopaedic surgeons. I would be more interested to see the incidence of mental health issues in psychs vs. oncologists, or doctors from another similarly traumatic profession where there's a high level of burnout.

As for psychs who manipulate their patients or violate sexual boundaries - unfortunately, psychology/psychiatry is more likely to attract predators than other medical specialties, because it gives a level of access to patients that other medical specialties don't. However, I have a hunch that if you looked at the mental health history of the minority of psychs that abuse their patients, you would find that they're less likely to have a mental health diagnosis than the psychs who don't abuse their patients. I would be interested to see if anyone has done that study, actually.