The correct study is one where some participants get an intervention to make them exercise more, and some are left alone. There's no ethical issue, but running such a study would be difficult and expensive.

Adjusting for BMI does not solve the problem, because there are many things besides BMI (unmeasured confounders) which affect both cancer risk and willingness to exercise.

And BMI is a pseudo science measure anyway. Its been proven to be a poor indicator in about every disease many times before.

the problem is that on noisy data, that adjustment of the variables isn't really as statistically clean as maths would say.

it's the same with red meat causes cancer studies. they say they remove smoking and lack of exercise and still wave the statistical significance.

it's very questionable how they removed the influence of these variables, since meat eaters do smoke way more, and do not exercise, and care about their health less.

Actually, I think noise is a red herring, and that worrying about noise (instead of worrying about unmeasured confounders and incorrect causal structures) is how epidemiology got into this mess in the first place.

Statistically significant does not prove anything, and it does not mean the difference would be large either.