Why do you think Pike's crack about running a cadet ship is anything other than a jab to get them to focus?I agree; Pike's ship/cruise has cadets aboard. Kirk's ship does not since it is on a 5 year mission (the concept of a 5YM didn't exist yet during the first pilot). In addition, Starfleet (not called that at the time) has specific training ships for cadets. As Pike says at the end of the episode:
What if he said: "What are we running here? A clown college?" That wouldn't mean he has actual circus clowns on Deck 4.
He was cracking the whip. "What are you? Cadets or officers? Get back to work, get us out of orbit."