I mean the character is not realistic, if by realistic we mean how a real starfleet officer would behave.
TOS Kirk gets away with far more bucking of authority than any real Navy officer would -- even of the Royal Navy of the 19th century, which Starfleet is arguably more than half-modeled on. But he's always acting within the authority of his command and making command decisions he can defend. By contrast:
You seem to be saying that the movies went downhill in part because of this kind of storyline with Kirk, stealing the Enterprise etc..
Yes, hijacking a ship and sabotaging another one Because Friends was a bad sign; we're meant to accept them as Kirk-like actions (though give Shatner due credit, he successfully sold them as "getting old, lovin' my friends and not givin' a f*ck"), but overall they were really a sign to me of sentimentality starting to suck the franchise down its deadly maw.
Not quite sure what you mean. I guess you're saying the entire story itself shouldn't have been told because it took the characters beyond believability. But that was the story, so playing within its context, if Kirk believes there's any chance at all of finding Spock and he can be saved, shouldn't he try to save him? It becomes a rescue mission for all points and purposes. I don't see any sentimentality in it. He unknowingly abandoned Spock, and is trying to make it right.
Essentially being called old and irrelevant by Starfleet (especially after rediscovering himself in TWOK) may motivate him to go further outside the lines on Spock's behalf that he normally would, but in TVH, we see he was more than willing to accept his punishment for it. Before, he always had a plausible explanation for his acts. This time, he offered no defense.
Further, by that logic, jeopardizing the mission in TVH to save Chekov is another sentimental moment, I guess. Should they consider the death of one good crewman compared to saving the world an acceptable loss and move on? Would that be more in character? More realistic? Essentially leaving a wounded fellow soldier who can be saved back behind the lines instead of trying to retrieve him?