It looks like the question surveyed was just "When somebody loses weight, where does it go?" The reason why FM physicians got this wrong is most likely just a framing issue. It's not a wrong answer; fats are broken down by beta-oxidation and the products are used in oxidative phosphorylation to generate ATP - which is then used for energy.

Honestly the question itself seems opaque and intentionally vague to generate surprising results. It's like asking a chemist "What happens to bonds during IR spectroscopy?" and then when they start talking about induced dipoles you say,"WRONG! The bonds stretch!!"