Firstly... I'm going to get this out... I am a lifelong Trekkie that started watching Star Trek with the original series weekend reruns that the good ole' Sci-Fi Channel aired. You know, before the hip and trendy name change.
Secondly I think they should quit with trying to play up the nostalgia factor. Star Trek is already an established cultural icon that everyone knows about. Just have Kirk be Kirk, Spock be Spock, Bones be Bones... ect... ect... besides, if you REALLY want to bring in a new generation... how much do you want to bet that they HAVE nostalgia related to Star Trek?
On to how Star Trek 3, not The Search For Spock of course, can be a fantastic sci-fi action adventure film that poses sociological questions without being a war mongering action thriller.
STORY: Adapt the "Whom Gods Destroy" storyline. Garth of Izar is a fantastic adversary that offers the opportunity for the crew to explore the far reaches of this new Trek universe. Perhaps add a twist that Garth was driven insane looking for some ancient legend in the area about gaining access to other parts of the brain... like from the powers of the Q.
Start with a cold open... akin to James Bond... set up that the crew has been out on the new Five Year Mission for either a year or two AT LEAST. Show that they are the Enterprise crew now. No more trials that are supposed to lead to them down the road. Maybe the cold open could allude to the tensions with the Klingons while the rest of the story is NOT about that like Thanos.
Maybe have the Enterprise report to a Starbase... hear about Garth of Izar's ship being "shot down" and then they get dispatched. I'm not going to go into a full plot, but have Kirk dealing with looking up to Garth for so long only to have seen him fall so far, get the REAL Bones/Kirk/Spock dynamic, get more background on Chekov, Sulu and Scotty with perhaps LESS of Uhura. Yes, I said it. But she and Spock have had PLENTY of development, let some others shine.
The point of Star Trek is "To Boldy Go Where No One Has Gone Before" so do that. Explore the human boundaries of greater access to the brain and ultimate knowledge.
Starfleet is meant to be NASA, not the Department of Defense.