Exercise has some huge physiological and psychological benefits [0]. It won’t immediately cure serious mental illnesses, but it definitely doesn’t hurt.
Anecdotally, my mental health is much better with a good diet and some physical exercise every week.
I love the fact that you call out exercise in its other forms. Physical labor is… exercise. Getting out and about and doing things can be exercise. It’s not just hitting a gym. I got in my best shape by avoiding the gym. There’s so much marketing around it, everyone trying to make money, when the reality is all you need is you. Calisthenics requires no upfront purchase. Only a time commitment. Lifting weights is cool too if you have them or want to buy them. Running is great too if you’re into that (or like the resulting “runners high”). I found sports, martial arts, and getting physical in my shop is plenty enough exercise to keep me in my current size jeans for the foreseeable future.
That’s true of diets, not exercise. Physical exercise is always measurable and always works. Whether you can do the exercise is another matter. It should be tailored to your body. It should be hard. It should hurt (in soreness, not in pain). I’ve seen people transform.