I think the weight of evidences suggest that the main factor is that protein is most satiating.
True, carbs - especially simpler carbs - are less satiating than fat, but protein seems to be the big hitter on satiation. This is likely the dominant reason why many earlier low-fat vs. low-carb studies appeared to show a metabolic benefit to low-carb; people are terrible at controlling their food intake, throwing the metabolic calculations off. Low-carb people, who are eating high-fat, end up consuming more protein... which is most satiating... resulting in lower calorie intakes. Low-fat diets tend to lead to high-carb, but not so often high protein... so people eat more.
Another nice things about the study in the OP: it carefully maintained constant protein intakes, as well as total calorie intake, so it's not a confounding factor.
Such a good study.
That is, in fact, a very tough question to answer.
In reality, I think we're finding that a lot of the advantages of moving a low-carb diet - such as increased insulin sensitivity - are really mostly an advantage when switching to low-carb after having spent years overeating in a normal or high-carb diet.
In other words, if your insulin sensitivity is reduced because you've been getting no exercise, and have been overeating, and have been sugar-bombing yourself between meals with sugary snacks and drinks... well, you'll get a nice benefit from switching to low-carb.
You'd also increase your insulin sensitivity by getting some exercise.
You'd also increase your insulin sensitivity by just moving to a calorie deficit for a while.
If you've burned out your insulin sensitivity, these are all good ideas!
But, what if you've been eating a healthy (or even slightly low) number of calories? If you're already lean and healthy, as many athletes are who actually have use for all those calories they eat?
Well, for them, there may well be no health benefit from low-carb, whatsoever. Maybe if there's no excess-carb "damage" to heal, then going low-carb doesn't help at all.
(And you totally can gain fat eating low-carb, if you keep overeating.)
So, if you're at healthy weight, and your metabolism is healthy, and you're not over-consuming, your body may run totally fine on many wildly different mixes of macronutrients. That seems to be what the science suggests: get your protein, get your essential fatty acids, and then you can largely fill up the rest of your calorie count with almost any mix you like. Until you start over-consuming, the body seems to do a really good job of taking care of things.