There's Nimbus III, a backwater world that the galaxy seemed to have forgotten about. Despite the hostage situation, the Romulans and Klingons never bothered to officially send anyone. Klaa only went because he thought he could gain some "street cred" by killing Kirk. Based on the response from other galactic powers, this doesn't seem like a mission a government would send a legendary, highly decorated Starfleet commander on, yet they send Kirk.
Here's the thing with that: I find it very hard to take such a poorly written script at face value for any kind of 'evidence' Starfleet was pissed off at Kirk. Here's the relevant quotes:
ADMIRAL BOB: I apologize for cancelling shore leave but look, we have a dangerous situation out on Nimbus Three.
KIRK: On 'The Planet of Galactic Peace?'
ADMIRAL BOB: The same. From what we can make out, a terrorist force has captured the only settlement and they've taken hostages. The Klingon, the Romulan and the Federation consuls. Now I know Enterprise is not exactly up to specs...
KIRK: With all due respect, the Enterprise is a disaster. There must be other ships in the Quadrant.
ADMIRAL BOB: Other ships, yes, but no experienced commanders. Captain, ...I need Jim Kirk. Your orders are to proceed to Nimbus Three, assess the situation and avoid confrontation if possible. Above all, however, all get those hostages back safely.
KIRK: Have the Klingons responded?
ADMIRAL BOB: No, but you can bet they will.
KIRK: Understood. Kirk out. Plot course to Nimbus Three, Mister Sulu.
So let's assume that Starfleet sent Kirk on this mission to try to intentionally get him killed (which is already a preposterous scenario, but I'm willing to play devil's advocate here.) Admiral Bob specifically wants Kirk to lead the mission, and Kirk states that his ship is not up to the task. So...why didn't either Bob or Kirk suggest just having Kirk transfer to a ship that was better suited to the task? The Excelsior was literally sitting right next to the Enterprise-A. Why didn't Kirk just take command of that ship? Also, this is a rescue mission into dangerous territory. Even if Admiral Bob had a hard-on for giving Kirk the mission, why didn't he send along a commando infiltration unit to do the actual rescue? Why, instead, does Admiral Bob allow Kirk to just use a bunch of senior citizens for such a delicate task? And why does Kirk just accept all this without batting an eye? And no experienced commanders? What happened to Captain Styles? Why would he not have been an 'experienced commander?'
The bottom line is that there is no logical reason for Kirk, his crew, or his disabled ship to take part in this mission, and yet with almost miniscule resistance, Kirk willingly does so, without questioning the logic of those orders. So neither Starfleet nor Kirk is acting at all rationally.
Then there's the Klingon peace talks after the explosion of Praxis. Starfleet brass sends Kirk with the full intent that he will be the patsy for assassinating the Klingon chancellor and be sentenced to death. Now of course instead of an execution, Kirk was sent to a prison, something I suspect the Starfleet conspirators didn't expect. And, yeah, sure, there were "rescue plans", but I suspect the conspirators knew that the UFP president wouldn't go through with it and create a diplomatic incident that could lead to an all out war.
First of all, it wasn't Starfleet's idea to send Kirk on the mission; it was Spock's, without Kirk's knowledge. And second, the conspirators' intent was to prevent the peace treaty, not to specifically make Kirk a patsy. If Starfleet had instead chosen, say, the aforementioned Captain Styles for the mission, then it would have been Styles and his ship that would have been framed, not Kirk.
I suspect this is true of any century; The military doesn't like anyone stepping on their peepee. Not only did Kirk step on it in TSFS (and other instances), he did a full on Riverdance on it.
And yet he also saved Earth, and presumably, the families of all those Starfleet commanders. So I don't see Starfleet being too pissed off at him.