I think anyone who fears for their mental health in a job like private music teaching should consider other options.
I don't expect this to be a popular opinion but taking up piano teaching after working in the NHS for a couple of decades I can hardly believe the joy in the autonomy of deciding for myself how I will do things, even to the extent of choosing my clients. Not a thing I could do in my previous career, however bad they smelled, however challenging their behaviour or hopeless the case.
I'm sorry - the stress levels are just not on the same graph.
It is impossible to say what the causes and scale of someones suffering may be and so any comparison is pointless. Everyone has a different tolerance to individual stressors. I was recently chatting to a fire fighter. I could not imagine coping with having to run into burning buildings. The fire fighter could not imagine coping as a teacher.
The lack of autonomy, respect, resources etc are undeniable stressors for NHS workers, but there are also benefits, perhaps the support of co-workers, regular income, hopefully a decent line manager and the patients come and go relatively quickly, care is by the nature of shift patterns a team effort, not a solo one, for example.
A lone teacher has no co-workers to turn to (hence forums like this, I guess). Pupils turn up every week (more or less) demanding full attention whilst sometimes giving nothing in return but expecting miracles. For some teachers that can lead to feelings of failure, of being in the wrong job, of not being good enough, of failing their pupil. There are no co-workers to share with when things don't work. Teaching is intense and many teachers become emotionally invested in their pupils' development as a musician. Some teachers have experienced bullying tactics from parents, or pupils. The isolation, the feeling of "having" to find an answer, with no one to turn to, the sense of ongoing failure are stressors which could lead to anxiety or depression. Different to the stresses in the NHS, but equally valid for the person experiencing them, and no less real.
I suppose you could also say that anyone working in the NHS must know to some extent what to expect before they apply, the media are full of stories about the stress, shortages, violence against staff, etc. (It amazes me that anyone would work there, but I am so glad that they do.) Similarly there are media stories about the stresses of being a school based teacher. Private tutors mostly get the "let's regulate them" kind of story. So, likewise, it is amazing that anyone applies to work in a school any more and why private tutoring must seem like a better, less stressful option. "Teachers to be" probably imagine tutoring will be "nice" (but enjoyably challenging) because their lessons were. Finding out that it is not always like that can be a bit of a shock (to say the least) as countless threads have shown.
Many people function in the world, as doctors, nurses, teachers, chefs, electricians... with mental health issues. It doesn't mean they should quit their jobs, or that they made the wrong career choice. You can love your job and still find aspects of it stressful to the point of breaking. It could be said that the more you care about your work, the easier it is to be affected by it.