Exercise pushes stuff around in your gut which is probably more directly related to the immune response than the possibility of eating raw meat.
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?