It's a legitimate answer in that, given any of those --- depression, anxiety, drug withdrawal --- you will probably quite literally feel better, more often, if you exercise than if you don't.
Of course, the same is true of someone who is experiencing no mental health problems. But although someone without mental health problems perhaps ought to exercise, and probably would enjoy both a significant improvement in mood and a reduction in stress levels; that person nonetheless doesn't need and hasn't been identified as someone who needs a significant improvement in mood or ability to cope with stress.
To my understanding, the reason that exercise is mentioned so often in the context of mental illness is that (a) it very, very likely to have a definite and subjectively detectable positive effect (unlike many potential medications), (b) it has very few potential negative side effects, (c) it may improve the body's ability to metabolize some drugs, (d) it is something that the patient can have control of, (e) it can add some structure to an otherwise unfocused life. Although it might have a "tipping point" effect for some people at some times, and some people may associate it with the end of an isolated episode of chronic depression, I don't think that it is often recommended (by doctors) in the hopes that it will 'cure' mental illness, per se. It can help treat it, and it can help treat confounding factors like lack of purpose, lack of success, lack of agency or autonomy, lack of social contact... et cetera.
I'm sorry that exercise doesn't provide any positive benefit for you. It's not entirely clear whether you're comparing sustained periods (on the order of weeks or months) of low exercise to periods of frequent exercise, or whether you're expecting to find a difference between a day on which you exercise and a day in which you don't. From my experience, I tend to think of it as a general metabolic effect. It's possible that it genuinely doesn't or won't help you; it's possible that you do exercise more than enough, and that you might (or might not) feel worse than you do now were you to cease exercising entirely.
I certainly share your frustration with people who deflect conversations in the direction of exercise as an answer to (let alone cure for) mental illness, most especially when they are proposing to "solve" a problem that they themselves have not experienced or have no substantive knowledge of.