I have a relative who survives on leaves and can't seem to shift any weight. I presume there is something wrong as she couldn't eat any less if she tried
She's either wilfully lying to you or is totally mis-counting her calories.
If you took any 20 stone person, locked them in a room and fed them 800 calories of food a day and water only to drink, they would lose weight - probably very rapidly for at least several weeks. Yes, even those with hormonal issues. Even with zero exercise.
It ALWAYS comes down to calories in.
It really doesn't matter what you do to reduce the calories that go in. Theres no right answer. Go low carb, Keto, fast, Slimming World, Weight Watchers, diet shakes, calorie count (accurately), Fast 800 - whatever works for you and you, as a person, finds maintainable.
They ALL have the same end goal - reducing the amount of calories we shove in our mouths.
The thing I find concerning about the injections is what happens when you stop them? All diets have a high failure rate because it takes willpower. But the ones that succeed just never actually end the diet - they learn willpower and keep doing the 16 hour fast, or stick to 1500 calories or keep eating in a SW friendly way for LIFE and that maintains their loss.
These injections appear to be a switch flick where you take them and you suddenly have willpower/no cravings. Great - except surely when you flick that switch off and stop injecting yourself, you'll be back to where you started?