Both Uhura and McCoy display shocking incompetence in their jobs. As a comms officer, Uhura should know basic Klingon, something that has been retconned in later iterations of comms officers. McCoy should have the medical knowledge to treat Klingons the same way he should be able to treat a lot of other aliens on Enterprise.

Scott and Chekov are slumming it on the Enterprise. Scott has been shown to be a great engineering marketing, so it doesn't make sense that he is still chief engineer on the Enterprise. Maybe Starfleet still doesn't trust Scott after Scott disabled the Excelsior, but Scott should be doing something bigger. Chekov was a first officer on another ship before coming back to the Enterprise as second officer. Chekov should be a captain by now.

Spock is going some high admiralty shit in this movie while being a captain, since he can't get promoted over Kirk. At this point, Spock should have transitioned to being an ambassador as part of this movie.

Kirk is really washed up career wise in this movie. He's only bring drug along because of Spock. He also is set up as a patsy for the bad admirals. It shows how low his star has fallen when he was used as a pawn rather than be an active participant in the politics of what is going on.