The problem with this article is that it's entirely hypothetical, save for one study that strongly indicates that obesity and poor health outcomes are linked (again, no causality).

If you wanted to argue that obesity is not necessarily associated with poor health in every individual, of course you'd have a good case. But most of the time we don't understand individual variation very well, so we give advice that would work well on average.

And losing weight does work well on average (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5497590/), so either the mediator variable not only affects weight, but also whether weight can be lost when you're in a study specifically designed to help you lose a certain amount of weight. (See citation 7 in the above review.)

So it's not just correlational. Here we have direct attempts to manipulate weight, and health benefits that increase as the targeted weight loss scales up.