I haven't read this entire thread, so forgive me if this has been stated. But, I see this Gary Mitchell reference come up all the time as describing Kirk as a cadet. It does not. Mitchell is describing Kirk as he was as an instructor at the academy: "Watch out for LT. KIRK, IN HIS CLASS you either think or sink." I have never served in the Navy, but I do recall reading in Tom Clancy's book about the submarine service that junior officers often spend a period of time back at the academy as instructors before they are given a command posting. Someone with more knowledge than me on that subject might clear that up. But in any case, the "stack of books with legs" was a Lt., not a cadet.
All true, but I still find it rather odd that in the film, Kirk needs an excuse to get on board the Enterprise. If he was already an officer, I don't see why he'd need to do that.
And the reason it's widely suspected that Kirk is a cadet in the film is not because of Gary, it's because of his black uniform.