I'm a Type 1 diabetic, so I have direct knowledge of exactly how much insulin goes into my body every day. (yesterday was 40.55 units, 24.44 as basal and the balance for meals) I've gotten some insight into the role insulin plays in weight loss because of this.
Years ago, I had gained a decent amount of weight over a lazy summer, simply from eating too much. I found the weight very difficult to get off, and mistakenly attributed it to insulin at the time. Looking back, the insulin was driving me to eat. Insulin lowers blood sugar levels, which makes you eat. I was staying fat because I was eating too much, and the excess insulin levels were making that condition harder to remedy.
I switched insulins about a year after that, part of the switch was changing the dosing. I pretty much melted, lost 35 lbs in six weeks without effort. Upon consideration, I realized I was no longer eating to excess, which drove the weight loss. The low insulin levels made this happen faster, but without a caloric deficit, I would not have lost weight.
I've experiment since with low carb/mod carb diets (I can't successfully pull off keto, always end up running my sugar low and having to eat carbs at some point). Low carbs works in the sense that it removes most of the bullshit snack foods that I like to eat. This lowers calories, now that I'm in a deficit, the low level of insulin allows me to lose weight quickly. The weeks where I overeat, even with low insulin levels, I don't lose weight.
My guess on most people who think they can all of a sudden lose weight without effort on low carb diets: they are not really accounting for calories accurately, and they are simply eating less, likely because they normally cheat with carbs. Remove carbs from the equation, cheating gets less likely, weight loss happens. This makes sense from a human perspective, and from a thermodynamics perspective.
Self reporting of calories is all but useless, individual success on dieting revolves around finding a system where you don't have to be strictly monitoring every morsel you eat or don't eat all day.