Remember in "The Enemy Within" how Spock put everything else on hold when the person looking like Kirk told him to concentrate on maintaining appearances and preserving the Captain's reputation? It would be inconsistent for him to do anything else in this case, either...
By Spock's logic, in both of these cases, having the real Kirk in command of the ship is less important than having a Kirk there, as long as the ship is in no obvious danger from the possible impostor.
Why is this? Would the ship indeed be thrown into chaos if it turned out the CO was a rotten apple, one way or another? This is not exactly Horatio Hornblower, where the crew consists of the dregs of the society and is ready to cut throats at the first sign of laxness from their bosses. But this is not exactly TNG, either: Kirk's crew in the early episodes is
undisciplined, argumentative and less than fully committed to its duties (say, "The Man Trap"). Protecting the apparent CO from impressions of weakness might indeed be crucial, and a priority job for the XO.