I just took it that Chekov felt he was going in over his head as chief engineer. It was one of thing for him to dabble in engineering, quite another to suddenly be put in charge of it, especially with the ship about to go into danger, IMO.
Yeah it was an in-joke. But the whole promote the navigator who only qualified from Starfleet a year ago to be chief engineer scenario was rather ludicrous. It shows Kirk up to be a hopeless adminstrator but I guess it's no worse than promoting a cadet to first officer.
It's so reminiscent of that scene in the Simpsons where Homer is put in charge of the submarine. 
