...when in fact they simply cut their Calorie intake in a way that they could sustain.
Absolutely, that's the key.
The other mistake people make, is that they starve themselves, thinking that they will lose weight. It's true, but only up to a certain point. The body is amazing and will do everything it can to survive. After about 6 days of 'starving', your body will go into a defensive mode in order to do what it has to, to protect the brain, the most vital organ.
The body will stop burning fat for energy and it will start to consume muscle in order to produce glucose to feed the brain. Fat is the last the thing the body will use as a source of energy.
This is the mistake I used to make. I could lose weight, by not eating or eating very little, but it was at the price of losing muscle mass. Now by keeping my calories at the level I need above my BMR, based only my level of physical activity and by increasing my protein intake, which builds and maintains muscle, I've been able to burn only fat.