This meta-analysis isn’t very convincing. Most of the studies included were primarily about other measures like IBS, COVID-related GI symptoms, or Fibromyalgia. Improving GI problems would be expected to improve mood.
The positive result is heavily driven by an outlier study on Fibromyalgia that has results that look a little too suspicious relative to the other studies.
> Improving GI problems would be expected to improve mood.
While improving GI problems might improve your mood, I don't think it is accurate to equate depression with "bad mood".
A friend of mine has IBS and while her GI problems are under control most of the time, she still gets hit hard by depressive episodes.