Maybe that's the point we're trying to make about Kirk's readiness to lead. If you think he'd falter the first time he had to sacrifice one of his crew because it was necessary for the greater good then what you are saying is that he isn't ready to be a leader. Remember Troi's (rather silly) command test? Kirk would fail.
Pretty sure I already acknowledged that I didn't think Kirk was ready for command. I'm pretty confident in that. As for sacrificing a crew member, my larger point is that Kirk needed to understand sacrifice-period. Something that his father did with question, that Robau did, and even Pike (to a certain degree). Kirk doesn't understand sacrifice. He will fight and fight and fight to keep a person from dying (Sulu, Spock, Pike) but he doesn't recognize why someone would sacrifice himself-there could be potential deeper psychological meaning there, but I'll leave that analysis for later.
Spock's act of self-sacrifice in TWoK is moving BUT it's also logical. He's qualified to effect repairs (more qualified than the trainees for sure) and his Vulcan physiology will allow him to last long enough to get the job done. I struggle to see the logic of Kirk's decision. The fact that there are no engineers even in the right part of the ship to kick the engines better could be argued to be a failure of Kirk's leadership ability in promoting an inexperienced 18-year old ensign, one year post-graduation, to be his chief engineer. And the decision to leave Scotty behind, well see Troi's command test above. ;-P
I don't see the logic in Kirk's decision beyond the fact that Kirk feels responsible. The damage to the Enterprise, the fact that so many of his crew have died because of his bad decisions, it is very much a movie moment of accepting responsibility and moving to make penance, or recompense, or whatever you would like to call it.
As for Chekov, I have my own theory about that, but it has nothing to do with my enjoyment of the film, so I'll leave that argument alone.
In the Kobayashi Maru test in TWoK, the captain survives and everybody else dies. THAT is the importance of the test although we know that both Kirks balk at accepting that as a necessary evil.
But of course there is a lot to love in the movies too. Plus it's not as if Kirk Prime doesn't make howlingly poor decisions on more than one occasion. Some of the plot lines and character arc work reasonably well. I just wish they were a bit more nuanced.
Forgive me, but I'm confused. I thought the point of the Kobayashi Maru was to face death by both the captain and crew? It isn't like captain is supposed to survive either, since they are faced with overwhelming odds and their ship is likely going to be destroyed as a result.
But, I do agree with your larger point that both Kirks don't like such a scenario. NuKirk simply faces the situation sooner than his Prime counterpart.