Obesity-What are we trying to do, cut calories or cut weight?

The obesity epidemic is monitored based on BMI, which is a measurement comparing weight with height. The body needs to consume a certain number of calories to be satiated and have enough energy to carry out it's daily tasks (this calorie consumption is guided by hunger). Now we are told that carbs have less calories per gram than fats (carbs have 4/g and fats 9/g) and thus we should base our diet on carbs and reduce fats to lose weight so that the same weight of food can be eaten whilst consuming less calories and this it is proposed will lead to weight loss.  However we can flip this calorie/weight argument to show that it might be better to eat more fats.  If we take 2 adults who need to consume 2000 calories a day to feel satiated. One gets their calories from fats and the other from carbs.  The person who satiates their calorie demand by consuming 2000 calories of fat will have consumed less weight (222g) than the person who has consumed the 2000 calories of carbs (500g). Thus the person eating the carb diet will be heavier than the person eating the fat and more obese according to the BMI method we are currently using to monitor the obesity epidemic. Thus consuming fat is the way we can consume the most calories and the least weight. With carbs you have a choice of eating half the amount as you would fat and consuming half the energy and being very hungry and or tired (i.e not meeting your energy needs to feel satiated) or you can consume your full calorie requirements and consume over twice the weight you would have done from consuming fat. I know this science is incomplete and doesn't take into account storage but nor does the current theory of calorie cutting or the BMI scale currently used to monitor obesity levels.