It would help if you would quote the relevant explanation that explains why their own summary—and I quote it again—does not say what I think it says.
The "Author's Summary" says:
> analysis of methodologically robust trials only shows a smaller effect in favour of exercise.
The summary does not seem to say that that the effect disappears. The summary seems to say that the effect is smaller when you only include "methodologically robust" studies, which I interpret to be synonymous with what you describe as "good" studies.
Can you please share what you're reading that says that the effect disappears entirely? Or perhaps you are quoting a different review than the one you linked?
Exercise as a treatment for depression does nothing for most people, and has only a tiny effect for the rest.