Genetics contributes significantly to the risk of obesity in various ways. It's highly heritable, but I've never seen anyone suggest it has considerable overlap with the genes associated with educational attainment or intelligence. That list by itself is just some measurements. You need a stat analysis of it of which would have to include a lot more data.
This doesn't really say much at all. It only suggest an association.
Conclusions
Future research should address these gaps to understand if the relationship between educational attainment and obesity may be causal, thus supporting education policy as a tool for obesity prevention.
Let's say they did get a study saying it is causal to some degree...this would probably be as effective as business ethics classes or high school PE. Taking classes is a poor tool for cultivating culture. And of course you're saying ANY higher education which obviously has more to do with other factors related to educational attainment.
Some highlights within the link:
Education may be associated with health via adult socioeconomic status (e.g. income and occupation4,7), but also via health literacy and health behaviors4,7-10 and sense of control and empowerment11.
For example, one study found that receiving dietary information as a way to reduce obesity was more effective among those with a higher education132. Alternatively, socioeconomic status may mediate the education-obesity relationship; most studies have not isolated the effects of education separately from other socioeconomic measures, including measures of income that may also be associated with obesity
We acknowledge several limitations to this study. Most importantly, the heterogeneity in the way articles measured education and obesity and reported measures of association hampered our ability to summarize across the studies beyond direction of association, especially since some of the measures of association may have reported total effects and others reported only direct effects. An additional issue when considering the studies that reported direct effects is that none adjusted for all possible mediators, and these analyses, which adjusted for mediators using more traditional approaches, could actually lead to additional sources of bias.