The story made sense to me.
A story can make sense, but the narrative, plotting, pacing, and execution can bring it all crashing down.
For example, what was the narrative purpose of that moment where Kirk asks McCoy what Spock would do if Kirk was in Spock's position? McCoy said he'd leave Kirk to die as if to imply that Spock would under no circumstance violate any rule even if it meant saving the life of someone important. There is no moment in this film that pays off this exchange nor is there any moment where Spock himself violates any kind of rule to save Kirk in the end. It's a long drawn out moment that never pays off.
And despite trying to give Uhura a "purpose" in this film, why even bother having her attempt to be diplomatic with the Klingons when it all turns into an action scene anyways? Doesn't this do more to show how useless she is since Kirk's plan to "run out shooting" was what they ended up doing anyways? All that scene does is show that Uhura sucks at her job. But at least she can use a knife and fire a Klingon weapon!...
And what was up with Kirk's "revenge is bad" speech in the end when every act of revenge helped ensure the characters got what they wanted? If none of these characters acted on revenge in any way, things would be much worse off. And this is the same guy who, after agreeing that he wouldn't outright kill Khan, still physically assaulted him even though he was unarmed and had already accepted his surrender.
Also, if Khan designed these torpedoes, why would he leave the explosive components inside of them? There was already a sensor dampening thing in it to prevent people from seeing what was inside of it, so why leave the explosive components if no one can see them? And if you know the torpedoes have your crew inside and you set up booby traps to detonate said torpedoes if they were tampered with, why did Khan not tell the crew how to properly disarm the torpedoes to ensure his crew's safety? He literally risked the lives of his own crew that the film beats us over the head that he cares so much about... for no reason.
And why Klingons? Shouldn't Starfleet be more angry with the Romulans for destroying seven of their ships, Vulcan and almost Earth after the last movie? What's the point of calling the ship VENGEANCE when it's not even fighting the people who were the cause of making Starfleet more military based? I think someone in Starfleet, knowing that Nero was from the future would want to take action against the Romulan Empire and ensure that Nero and all those like him wouldn't take such action against the Federation again. Unlike the Klingons who have, well, just atacked us on numerous occasions, the last film established taht we are not at peace with the Romulan empire and if you want to bring in other TOS elements, there are officers in Starfleet who have a very open grudge against the Romulans over the Earth/Romulan war almost a century ago. To assume no one would care about the Romulans because Nero was from the future is like assuming every single person in Star Trek is rational and understanding. Even Star Trek with all it's Utopia preachiness isn't all 'that' perfect.
Yeah. The story can make sense, but that's all it has going for it.