The sense that I am getting at this point in the book (a little over halfway) is that Spock is reluctant to take any official action to relieve McCoy of command out of concern for his Starfleet career.
While on the surface this might seem like an illogical concern, it has been made very clear that logic is a country that some of the Admirals in Starfleet have never heard of let alone spent any time in.
The best example is when the admiral who contacts the ship tells McCoy in the same message to both pull personnel from working on trying to improve the translation matrix for the aliens language AND to hurry up and establish clearer communications with those same aliens.
In other words...
"Take off your hat. Raise your right hand. Place your left hand here. Take off your hat."
As for Spock not taking shots at McCoy when he has the opportunity, well obviously everyone has their own take on matters but to me such behavior would have been horribly out of character for Spock.
First of all (to me at least) he and McCoy actually respect each other, even like each other, but they are both stubborn and argumentative in their own way and take great delight in using the other to sharpen their wits on.
Even so Spock's commitment is to C'thia, to Armien'tu and to his role as First Officer and the good of the ship. Now if he truly believed McCoy to be incapable of serving even temporarily in the role into which he's been thrust I feel quite certain he would have relieved the doctor of command in a Thrasian heartbeat. Clearly he does not. However he knows that the doctor is out of his element and there is no logic in making an already tenous situation worse.
Now after the danger is past and everyone is back where they belong? Then I suspect there may be some very dry needling on Spock's part.
As for the book not giving an insight into McCoy I disagree. He spents quite a bit of time ruminating on the times that he's blithely given Kirk advice, not really realizing how often his advice might be at odds with the realities of command that Kirk was dealing with.
As for his approach to the Klingons, while it might be informed by McCoy asking himself WWJKD, it is also clearly informed by his knowledge of Klingon psychology. In fact I think you can add McCoy not knowing anything about Klingon physiology to the list of sins committed by TUC.
The most jarring bit for many people not conversant even in passing with the earlier iteration of the novels is the Klingons and their urgent search for a particular substance on the planet.
Basically at this time some authors had established that without a kind of drug Klingons were incapable of controlling their violent urges, and the crew of this ship had run out some time ago. It's like a less intense version of ketracel white basically.