It’s truish. Broadly, insulin impairs the ability to access fat. It doesn’t actually shut it off completely. Aside from anything else, you are literally never without insulin in your bloodstream unless you’re a type one diabetic. If ‘can’t burn fat with insulin’ were true, no one would lose fat ever.
From an anecdotal standpoint, I’ve lost fat while taking enormous doses of exogenous insulin, in addition to the fairly huge amount my body produces all by itself.
And generally, a lot of the accepted knowledge is like that - biochemistry is ridiculously complex, and any simplified guideline should be taken as a rule of thumb, not a law of the universe. Protein spikes insulin less than carbs, but not all protein has the same effect on insulin release. And so on.
It’s possible to die from malnutrition while obese, probably, but you’d need some kind of really rare metabolic disorder to actually starve to death (which is to say, death coming from cannibalizing vital tissues).