See, that's the root of your mistake. You're basing your argument on what you know after the fact, rather than putting yourself in Kirk's head and considering what he knew, and more importantly what he did not know, at that point in the film. You're saying that Kirk should've perceived Khan as a threat so heinous that killing him was the only recourse, but he could not possibly have known that yet. He certainly had reason to mistrust Khan's intentions and to keep a close eye on him, but also to consider him a tenuous ally against the greater threat of Marcus. He had absolutely no reason to default to killing Khan at that point. It would have been immoral and probably illegal for him to kill Khan before Khan had posed a clear and present danger that justified escalating to that kind of response. You can't fairly condemn a man for failing to act on information that could not have been available to him yet.
Yeah this is all true. It's also worth noting that in the previous movie, at the time that Kirk tried to mutiny against Spock, Kirk was not in possession of the additional information from Spock Prime nor the formula for transwarp beaming. Ergo, Kirk's intention to pursue Nero to take him on was highly likely to fail. Spock's decision was also wrong of course - the correct course would have been to contact Starfleet Command on Earth, Andor, and Tellar, notify them of Pike's capture and ask a ship closer to the fleet to get a message to them. Starfleet could have changed the codes on Earth's defences and at warp 8, the fleet could easily have beaten Nero to Earth.
Obviously, there is no guarantee that the fleet would have beaten Nero without the extra intel from Spock Prime but I only use it as an example that while hindsight can condemn or condone, the reverse is also true!