I'm no military man, but I understood well enough what was going on. It, like most of the movie, just didn't make any sense.
Why would the brand new flagship of the fleet be crewed by raw graduates who can't follow a checklist to launch her properly? Why would the captain promote a stowaway cadet on academic probation to second in command when he has a bridge full of legit officers? And then why send two of his bridge officers—including the guy he just promoted to second in command—on what is essentially a suicide mission to stop the drill?
My biggest problem, tho, is that Kirk's meteoric rise could have been handled much more smoothly and believably if they'd made any effort....
Kirk cheats at the Kobayashi Maru test and gives an impassioned defense about how his job was to win, not to participate in Spock's psych experiment. Kirk is graduated and commissioned as a lieutenant. Nero makes his move, and part of the fleet takes off for Vulcan—and is never heard from again. Starfleet recommissions some older ships and promotes new graduates to crew them. Kirk is assigned to the USS Finifter as a lieutenant commander.
The second part of the fleet goes to Vulcan more warily and encounters Nero, who destroys some of them, including the Finifter. Kirk is a survivor who gets beamed aboard the damaged and short-handed Enterprise. Pike, impressed with Kirk's bravery and leadership, and short-handed himself, promotes Kirk to commander and assigns him as first officer to make Spock the captain.
Now, this makes Kirk an impetuous hero instead of a rebellious screw-up, but guess what—that's what Kirk is supposed to be. Let his impetuousness show thru in the midst of his heroism instead of showing him repeatedly making career-limiting moves.