To try and save on the budget a tad, but also delve into some world-building that the previous films somewhat glossed over, I propose the following:
The five year mission of the Enterprise and her crew has come to an end. The Federation flagship has one last assignment before it is expected to return to the heart of Federation space, where she will take on more general and traditional duties.
That assignment is to oversee and finalise the accession of a planet along the Klingon-Federation neutral zone into the Federation, a move to which the Klingons have not responded well; the planet is technically within the neutral zone. Thus the Enterprise's presence shall be as much deterrent as it is measure of goodwill.
So from this you have:
A) Political intrigue. An angle through which we can be allowed some look at Star Trek politics while also being far enough away from the core that we can keep things simple. Federation feels that as the border was drawn up with no knowledge on the part of those placed within it, if those people wish to join the Federation of their own free will, that is entirely their right. The Klingons see it merely as a territory and power grab by the Federation, to gain the resources of the planet (let's say it's got a lot of dilithium or something; just remind the audience that it's starship fuel, and the interest becomes easily explained).
B) Room for debate. Literally, scenes can easily be constructed out of people speaking over dinner tables and in conference rooms, minimising set piece moments until necessary. Should hopefully save on the budget, which the current Trek films desperately need to do in order to better turn a profit.
C) A chance to explain... changes, and drive character drama. Chekov can have gotten off the Enterprise for a new assignment, and the crew have wished him well, wherever he may be. Jaylah could be picked up from a starbase they stop at during the start of the movie; the indication being that she graduated from Starfleet Academy in good time - Kirk can be slightly impressed but still boast that he made it to Captain in three years, not merely graduate. With it uncertain if the crew will stay together once they're back home - the Enterprise now having some of the most experienced people in the fleet, and Starfleet may want to spread that experience around, as they already have with Chekov - you can give people like Uhura and Spock a push to get things resolved. Kirk can contemplate his navels about the possibility of being responsible for a new crew, talking to Bones about it as the good doctor is trying to figure out how to best compile all the medical knowledge he's gathered over five years so that it's applicable for the scores of doctors that will follow him - considering the next generation to come. Sulu can form a dynamic with Scotty and Jaylah I guess, where Jaylah is kind of the bridge between the two men; a sparring partner for Sulu's 'fencing', interested in engineering, and in general having a curiosity about the ship - what two members of the crew would know it better?
D) Mystery. Similarly to Trek VI I suppose, but needless to say, the accession of the planet would not go smoothly. Something, somewhere would explode, with fingers being pointed all over the place as to the culprit. It can look like the Federation did it, the Klingons accuse they would seek to take over the planet, get brought in as a 'mediator' while investigations takes place. Or it could look like the Klingons did it, and so Kirk is pressured into defending the planet's sovereignty and proving that the Federation would be a reliable ally - that's what they're there for, right? - even if it means triggering a war. Whether or not there'd be a third party who actually did it, I'm not sure; having the Federation somehow be behind it feels too obvious a twist, and easy to mess up, while the third party option would require more exposition and explanation. Romulans maybe, seeking to light the powder keg the two other superpowers have been sitting on since ID, and there'd be some familiarity to build off of, but it does feel sort of cliché for this franchise and would require some good foreshadowing so as not to come off as unnecessary fanservice.
Admittedly, that's a plan for a single film, rather than overall output, but I feel it lays the foundations for what can go ahead: The Enterprise's shift in status quo is addressed, Jaylah can be integrated into the crew as a consistent character rather than have someone new made up to fill Chekov's void, and it may perhaps ease audiences down into a place where some of the weirder but slower Trek stuff can subsequently occur.