The problem with this particular issue has several somewhat contradictory components.
One--the use of "military ranks" causes many to assume the ranks work as they do today.
Two--what little we've seen of the Academy onscreen (and know of the background of the characters at the Academy onscreen) provides murky glimpses of its workings. I suspect this is quite deliberate, but it leads to arguments like the one going on right now.
Three--a minor component to the argument is the unwillingness of those who want to see a more "realistic" portrayal of rank to accept precedence from prior Trek (complicated by the alternate timeline).
Like warp speed, time travel mechanisms and technology, ranks in Starfleet are, and have been, in service of the story first and consistency second. This is no different. It doesn't mean everyone has to be pleased about it, but it does mean that expecting a more "realistic" rise to Captain by Kirk is, well, at the risk of a bad pun, unrealistic.
The stated goal of the filmmakers was to create a story that portrays how the seven main characters came to be together on the Enterprise as the crew we knew best. That means Captain Kirk (not Lt. or Lt. Cmdr. or whatever), Mr. Spock (not Captain), Lt. Uhura, Dr. McCoy, Chief Engineer Scott (not 2nd or 5th or whatever in the engineering section) and so on. Anyone who thought this film would end with Kirk as anything BUT Captain of the Enterprise should have known better. No matter how interesting the backstory of the Republic or the Farragut or the "stack of books with legs" or...might be to some long-time fans, it would not have been of interest to the general public. So none of that was EVER going to be in the movie.
As to the "too swift rise" to Captain, precedent has already been cited (Treks II-IV). Moreover, the fact is it is entirely unclear exactly how the Academy works. As such, it could well be that the "command track" is established early on, for those with potential, and they graduate (after the Maru test) with a higher rank than non-command path graduates. Think how the services worked 300 years ago and is it really so difficult to imagine that substantial changes could occur 300 years from now? And, in the end, it doesn't matter. The "needs of the story outweigh the needs of satisfying every single expectation of individuals in the audience."

I guarantee that an exceedingly small number of people bothered to question Kirk's "too swift rise" in the movie. They simply expected to see Kirk end up as "captain of the Enterprise"--which is what happened.
Debating the merits of how it happened can be an entertaining diversion, but it is folly to expect how it happened to reflect "reality" all that much. There is no precedent for it before, why should it be different now?
To put it in non-Trek terms--I'm an historian and one of my research fields is historical feature films. I have colleagues who won't watch such films because any "inaccuracy" drives them nuts. However, I have a different attitude towards such films--those "inaccuracies" can lead to very useful discussions (and such films have a profound influence on how the general public interprets history, so I prefer to know about it than pretend it's not worth knowing). If I let every "unrealistic" moment in an historical feature film distract me, I could not watch them. That doesn't mean there aren't any that are too egregious to ignore (I'd hardly be doing my students any favours if I had that attitude) but it does mean I make allowances from time to time in the interest of having an interesting narrative.
To come back to the latest Trek film, that is how I view Kirk's promotion at the end. It allows the filmmakers to do what they want (show Kirk become captain) in a way that is satisfactory to the majority of the audience (i.e. without excessive back story baggage). Could it have been better? Of course (a little blurb saying "two years later" or even "4 years later" would have been how I would have done it). But it is not a "deal breaker" for me as the story doesn't depend on the precise mechanism of how Kirk became captain. It simply shows the journey.