Yes, Chekov is young, inexperienced and somewhat hot-headed (see Trouble with Tribbles) but his sheer terror on finding the dead scientist just isn't very believable.
While Kirk would no doubt resist any suggestion that he's no longer fit for command, so much of his character is built around sacrificing himself heroically for his ship that he would have readily stepped down with a little convincing that he had a problem. He certainly wouldn't have been relieved involuntarily after a long and rather dull competency hearing in which he makes a fool of himself.
Commodore Stocker is the most unbelievable character of all. He might be a chair-bound paper pusher, as Kirk says, but it is simply not at all credible that any Starfleet officer, much less one of such high rank, would be so utterly boneheaded as to violate the Romulan Neutral Zone just to get to his new job. (Then again, Captain Schettino was somehow hired by Costa Cruises, so maybe ship staffing fiascoes by upper management still happen in the 23rd century).
Yet this does set the episode up for one of the more satisfying endings; it's more common to see a promising Trek episode end with a whimper. It was fun to see Kirk reuse his classic Corbomite bluff, this time in clever combination with a Starfleet code known to have been broken by the Romulans, but which fact the Romulans apparently didn't know yet.
And it's been kinda fun to compare the "aged" characters with the actors in real-life old age. They look nothing alike.