How would you tell the difference whether exercise reduced your anxiety, or the fact that you were exercising was a sign that your mental health was better? I definitely feel more like exercising when I'm not anxious or depressed. Almost nobody is going to bother with exercise if they feel hopeless.

Besides, anxiety is not like cancer, where the tumor is either there or not. There are probably biological inputs, but psychology is obviously important. If you can get someone to believe they are not anxious, then they're not anxious. So, arguably, a placebo effect is the only known cure.