The first seven broadcast episodes of ST:TOS included one episode with a shapeshifter, two with faux Kirks, one in which everyone's personality changed, and two with omnipotent beings (one right after the other). And one in which none of those elements are present, but certain new characters nonetheless turn out to be not quite what they seem.
After all that, you'd think that anytime anyone did something strange on the ship, everyone's first thought would be “I bet they aren't really who we think they are!” Especially if it was Kirk who was doing something strange.
And Kirk could start to take advantage of it. He could behave badly in all sorts of ways, and nobody would be surprised; they would just assume it was another double.
“I did what? No, no, that wasn't me. Must be another doppelganger loose on the ship. Computer, sound the Doppelganger Alert!”