As far as anything beyond a trained medical proffesional gives you, its case by case basis to if it works or not. As the NHS puts it for Yoga "yoga teacher are not medical proffesionals and are not trained or qualified to diagnose or prescribe treatment to medical issues, but some of the exercises they do may or may not help (insert what ever joint pain you have here)" that in short means, it may or may not, until you ask a GP or run it through a GP or specilist its not a yes or no. (they cant actually tell you it will unless they have a medical proffesional backing them, or specfically run it through who ever is heading your treatment)
A martial arts teacher seems to be a part time, secondment to most people, so nothing says they cannot be qualfiied and work proffesional as what ever medical proffesion full time, but the large amount dont seem to be. Is the issue of martial arts being seperate from modern sports knowledge and education still as big as i think it used to be? Because if you say classes on how to be a personal trainer you tend to learn how to work around some medical conditions, or adapt training to them, or general aches and pains. (not to identify or treat them)
Id exepct the baseline quality of a martial arts teacher to be that of a trained personal trainer, can work around medical conditions and knows about them enough to work around them.
As far as i know at least here, the only medical proffesional that can diagnose and manage mental health issues is a psychatrist, so a psychatrist would head your care plan and vet and review anything you are doing and reccomend things based on their assesments. In some cases i think a Psychologist can, although that seems to be a largely treatment and research position. Every position has a list of what they can and cannot do and what autonomy they have.
So working off the teacher being a trained and qualfiied counciler and registered to work in that capaicty, they could potetionally mix martial arts things in with treatment. There is a needed level of proffesional conduct though, so if its done in that(proffesional) capaicty it needs the required proffesionalism alongside it. (no idea what it is, i just know its diffrent, very complciated and can get very legally complicated)
Addendum: Therapy on the NHS is at least free for me, so its a lot cheaper than going to Martial arts lessons.

Depends how you define therapy, some of its free some of it isnt, some of it can be discounted.