Why would you go run (hunt) if there is still stuff (food) in your gut.
Hunting isn't like going to the store, success isn't guaranteed.
Ketogenesis requires over a week of carb restriction IIRC. I think fasting yields energy because the body increases various stress hormones to push you to find food so you don't starve.
Have you ever tried to focus on a hard problem at work after a big lunch?