What people? As far as we were shown, there were only two people at the base, Scotty and the midget. Not much of a rescue team. They also didn't know Kirk was coming. They expected someone to be arriving with supplies at some point, but Spock apparently didn't inform them of the incoming guest or even contact the base at all. Eventually (and a rather fast eventually) Kirk would run out of food and/or air and have to leave the wombie space pod. And without the aid of old Spock, Kirk would have been big red monster chow or possibly dead of exposure.
Yeah, the rank jump is a wall banger but it is "justified", if still completely illogical and nonsensical, by being orders of Captain Pike.