As someone who struggles to manage mental illness, yeah people have become really knee jerk opposed to lifestyle change approaches to managing mental health. The research is really unambiguous. In DBT they teach you to exercise as part of the PLEASE skill for attending to your physical wellbeing, because its hard to be mentally alright if you feel like crap physically
Moving won't fix your mental health, but your mental health is unlikely to improve if you dont go physically move, its a really important part of taking care of your body
That being said, how you deliver a message has a huge bearing on whether its receieved, and the reason your message is one people are guarded against is because people constantly talk down to mentally ill folks and tell them how to "solve" their issues from a place of presumption. As a part of that, people frequently present exercise as a cure, and as someone who exercises every day it very much is not. Empathizing with that is worthwhile, as invalidating people also promotes poor mental health outcomes :)