I think it's an awesome idea and definitely the way to go if things continue in the way of a movie series. It might be a good idea to mine the original stories the way you would the Arthurian legends. Like when Mallory pulled together all the existing tales from the previous centuries and wove them into a single narrative.
There are a couple of strong arcs. For Kirk WNMHGB - Gary Mitchell's transformation and what it did to him, The Enemy Within and what that taught him about his nature and sources of strength, CotEoF and the loss of Edith, The Doomsday Machine and the loss of Deckerd, Arena and Kirk's ability to be merciful, A Taste of Armageddon and his ability to come up with radical solutions. That arc would all be about Kirk as commander, the things that isolate him and forge him into Starfleet's most legendary captain.
Then there's Kirk and Spock's friendship/ Spock's arc. That'd be This Side of Paradise and Spock's dual nature, Devil in the Dark and his tremendous compassion, Space Seed and the confrontation with Khan (foreshadowing TWOK, though there'd really be no need to remake that), Amok Time and Journey to Babel (or at least some story illuminating Spock's troubled relationship with his parents).
Either of those could be done by pulling parts of episodes, making references to some events, and weaving other major events from the episodes into a narrative. Looking at it, there's maybe a trilogy in there with the friendship as the main emotional arc and Kirk and Spock's development into extraordinary figures as the main "adventure" plotline.
I'd kill to see a more involved Mirror, Mirror - but that has to have some built up investment in the characters, and should connect back into the main universe much more strongly.