Since the original post indicated that the BBC program was based on misunderstanding the Atkins mechanism, the topic cannot be discussed without correcting their misconception. The study my post referred to disproved conslusively the idea that Atkins works only because Atkins dieters eat fewer calories. This is nothing more than the death throws of the adherents to obsolete theories.
The consuming more energy is supposed to happen because there is a loss of calories in converting fatty acids to ketones. So it might seem like you're overeating while you're not getting enough calories.
(1) "Energy in vs. energy expended" is the beginning and end of dieting. (2) Low fat diets are healthier than diets containing a moderate amount of fat.
Seems to be more a matter of relativity. I think "low fat" diets are really more lower fat, meaning don't gorge yourself on fried foods and sausage and cheese at every meal (basically ... lower fat than what people who are overfat would eat), and that leaner meats, poly and mono unsaturated fats, nut fats, vegetable fats, etc. are perfectly healthful and should be an essential part of one's diet. I've seen "low fat" diet suggestions that actually recommend a higher percentage of fat than protein. Typically, 60% calories from complex carbs, 25% from fat, 15% from protein. Sorry to deviate from the apparent point of the thread, but I think as a society we have too much of an attitude of "a little: good, a lot: GREAT" as well as "a lot: bad, NONE: good." A corollary might be "a part of the whole: bad, ALL parts of the whole: bad," i.e. refined flour and simple sugars: BAD, ALL carbs: BAD.