That is correct. Your Body mostly needs good protein sources (there’s no such thing as too much protein intake except if liver/kidney diseases exist already) since it can only reuse part of those in the body, not synthesize all necessary forms of it. Everything else (fat and carbohydrates) is purely energy. Sugar, starches and anything with sugar is just carbs to the body in different forms. The body can synthesize those as needed, whatever of both is deficient. Your body most likely runs a lot better on fat, according to anyone who tried.
Vegetables do provide some nutrients and buffering, yet they’re not an absolutely necessary main source IF you have access to good quality meat, eggs, fish, etc. Those things have a very high micronutritiants-content, as long as they are of good quality.
On Rabbit poisoning, looked it up. That is based on an absolutely purely protein-based diet with little to no intake of fat and carbs, as far as I understand that right now. I did not state that eating purely protein is okay, just that the body can make up for deficiencies in fat or carbs as needed. So eating nothing but chicken meat without any fat or carbs would likely cause rabbit poisoning symptoms as well. Which is, again, not what I stated.