When Spock turned command over to Kirk, the responsibility for everyone and everything on the ship and the ship itself transferred to Kirk, too. It's the nature of command. You can't split responsibility. It rests alone with the ship's captain. Spock went back to the science station on the bridge and filled his old roles as science officer and first officer. I did say Spock certainly knew the ship as well as Kirk. Since it apparently became a training ship, it's unlikely it would've had any new equipment on it Kirk wouldn't be familiar with (unlike in TMP, where the refit with new technology meant Kirk had to catch up). I'll grant you that Vulcan endurance may have given Spock an advantage in the situation, but that doesn't mean the thought of getting the mains back on line the way Spock did it couldn't have crossed Kirk's mind to at least try. Instead he stood there on the bridge and looked helpless.