The title of the post and article are both misleading. The study was only done with 8 year olds. The authors note that their conclusions may not apply to kids in general. Older kids are more concerned with appearance, and more likely to be self conscious, or to be bullies. I also feel like very high BMIS are probably a lot rarer in younger kinds who have had less time to put on the weight, and who usually have lots of energy. So if there is a causal effect, it seems it would be the smallest at the beginning of elementary school.

The actual paper also does state that higher BMI is associated with depression. And that "We found inconsistent evidence that a child’s BMI affected their depressive and ADHD symptoms [because it could be that the mental health issues are causing the increased BMI]". And "Our results suggest that interventions designed to reduce child obesity are unlikely to make big improvements in child mental health. On the other hand, policies which target social and environmental factors linked to higher body weights, and which target poor child mental health directly, may be more beneficial."

Basically, it sounds like the study is saying "Heavier kids are more likely to be depressed. We aren't really sure whether helping lose weight will help their depression. But we do know that helping them with their depression directly has good results, and could also help them lose weight, so give that a try." Which is an interesting and helpful conclusion.