Something about this article seems a bit off to me. I mean, sure, fat obviously (to anyone with a background in biology) is broken down and ultimately its carbon is exhaled as CO2. But to say that it's used as energy isn't wrong – what do these people think energy is? It's not just some mysterious molecule that turns your fat invisible; rather, energy is stored in molecular bonds, and released when large molecules (such as glucose, or fat molecules when not enough glucose is available) are broken down into smaller molecules, CO2 (carbon dioxide) and H2O (oxygen). Furthermore, fat can't be converted to CO2 and H2O without a release of energy (heat being one form of this). Doctors aren't stupid; this article is just misleading.
I think some doctors actually are stupid or at least narcissistic. But I do see your point here, also want to point out the absurdity of the graph chart they included. However the CO2 is released with activity then exhaled. That is to say we do not lose weight just by breathing alone. (would that we could!) So the energy has to be released as you said, first with activity then exhaled as carbon dioxide.