Kirk's hubris, his zeal to use the crisis as an excuse to get his ship back, has resulted in an unready ship being pushed into service, and two people have died as a result. And Kirk has to live with the consequences of his actions.
Why is Kirk responsible here when the sequence of events in the movie points to a mistake on the crew?
From a legal standpoint, he probably wouldn't be deemed responsible; after all, it was an emergency situation, and it's understood that the lives of military personnel are at risk in any crisis. But I was speaking more from an emotional and dramatic perspective. If Kirk hadn't forced the crew to get ready in a hurry, then they would've been more careful and fewer mistakes would've been made, and the accident wouldn't have happened. Its intention as a dramatic beat in the story was to force Kirk to face the cost of what he was doing -- that there was more at stake than just his own craving to get command back, and that being a commander meant being aware of your responsibility to the people under your command, the people whose lives are in your hands. It was part of Kirk's journey in the film, his arc from covetously and selfishly craving command to actually earning
that command again.