Kirk should have been a commander at most at the end of the first film (or Lt-commander if they had kept Spock at that rank too) so he could have spent a stint as a naughty first officer instead of an apallingly bad, dangerous starship captain. His progression in the movies is good, just not the unnecessary rank hike.
To be fair, the reason for that was mandated by the studio. The suits weren't entirely convinced that 'Star Trek' would be successful enough to spawn sequels, so they wanted to hedge their bets and insisted the movie end with all the TOS characters in their regular places heading 'out there'. This necessitated unrealistically bumping Kirk up in rank.
'Into Darkness' kind of tries to "fix" the situation by busting Kirk back down a peg or two, but it isn't a perfect fix. IMO in an ideal world, the 2009 movie would have suggested the crew all going off to other assignments at the end (so as to imply they each have some kind of further career before Enterprise), with STID then finding an excuse to bring them all back together again and finally sealing the deal of them all staying
together at the end.
For me, they could have promoted Kirk to first officer on a another ship with him cheekily announcing that some day soon he'll be back on the Enterprise or possibly even FF a couple of years in a scene at the end of the credits and show Kirk striding onto the bridge, newly promoted. He'd still be green but not unrealistically green.
Similarly, they could have promoted a more senior engineer to the position of chief but used Chekov as the liaison with the bridge to deliver all the dialogue.
I don't expect Star Trek to be perfect
but if they dip the level of realism and scientific plausibility too low it becomes more of a children's franchise and I think that would be a shame.