Love both nu-trek movies. Reading the laundry list of issues the OP stated back on page one, the only one that I feel holds merit is Kirk's rapid promotion. If you do something heroic you get a medal and maybe a promotion one step ahead. Your heroism doesn't all of a sudden give you the insight or experience to handle the administrative requirements of being the CO of a starship like the Enterprise. I'm talking about managing several department heads and fully understanding the assets and enablers that you control as well as the requirements and expectations that come with the job.
That being said, I expect to see CAPT Kirk in command of the Enterpise on the big screen so it obviously wasn't a deal breaker but definitely something that could have been handled better.