Just as you said, the gist of the argument, for this and all of the diets is protein:
A recent review by Hall  states, “There has never been an inpatient controlled feeding study testing the effects of isocaloric diets with equal protein that has reported significantly increased energy expenditure or greater loss of body fat with lower carbohydrate diets.” In light of this and the previously discussed research, the ‘special effects’ of LCD and KD are not due to their alleged metabolic advantage, but their higher protein content.
And more to your point
If there is any advantage to KD over non-KD for fat loss, it is potentially in the realm of appetite regulation. Under non-calorically restricted conditions, KD has consistently resulted in body fat and/or body weight reduction [65, 66, 67, 68, 69]. This occurs via spontaneous energy intake reduction, which could be due to increased satiety through a suppression of ghrelin production . Moreover, KD has demonstrated hunger-suppressive effects independent of protein content.
The collective evidence in this vein invalidates the carbohydrate-insulin hypothesis of obesity. However, ketogenic diets have shown appetite-suppressing potential exemplified by spontaneous caloric intake reductions in subjects on ketogenic diets without purposeful caloric restriction.
And isn't this the point?
In a 4-week crossover design, Johnstone et al.  found that a KD consumed ad libitum (without purposeful caloric restriction) resulted in an energy intake reduction of 294 kcal/day.
I understand the argument that the weight is lost because ketoers eat less, but that is the entire point, I am burning antique Krispy Kreme's and don't need to continually stuff my face. My body is getting what it needs from fat stores and not bothering me.
I don't read this as a condemnation or anything of keto, though the keto section is twice as long as any other and bases almost all conclusions off of Hall, but rather a platform for why they think it is still CICO and nothing more.
They follow up that logic in the strengths and limitations section:
Excludes/minimizes high-CHO foods which can be nutrient dense and disease-preventive. Can compromise high-intensity training output. Has not shown superior effects on body composition compared to non-KD when protein and kcals are matched. Dietary extremes can challenge long-term adherence.
Nothing untrue there except they still don't get that the purpose is to use body fat for energy, matching intake kcals with other diets would obviously miss the point of the diet entirely.
I thought the focus on protein was odd throughout, everything is brought back to copious amounts of protein, going so far as to recommend 4x the current reccomendation:
Increasing dietary protein to levels significantly beyond current recommendations for athletic populations may result in improved body composition. Higher protein intakes (2.3–3.1 g/kg FFM) may be required to maximize muscle retention in lean, resistance-trained subjects under hypocaloric conditions. Emerging research on very high protein intakes (>3 g/kg) has demonstrated that the known thermic, satiating, and LM-preserving effects of dietary protein might be amplified in resistance-training subjects.
And then you read the Competing Interests section where everyone involved was or is paid by a "nutrition" or supplement company. Hell some are literally on the board of protein supplement companies.