The rabbit thing is real... That is why you do have to eat it all including the organs. It is the same with quite a few species of very lean wild game. We just cannot survive without calories from fat or replacements for it.
https://www.raising-rabbits.com/rabbit-starvation.html
You can get fat from rabbits.
You just need to eat the fat the rabbit offers.
You can get quite a bit of nutrition from the partially digested stomach contents as well.