What is the scope of these "core" powers? During times of war, the Commander in Chief directs military maneuvers etc. in active theatres of combat. At all times he directs readiness and training.

I wouldn't say that the history and tradition of the office is to have Delta Force manufacture artistic pottery cups for children to drink milk from while at school. That action in that time and place would not seem to fall within the role of Commander in Chief (regardless of his intent / subjective belief that the use of fine porcelain may provide some tactical advantage).