Sirgod's comment had a thought pop into my head, that is that we really don't know a lot of the backstory of the various characters, when you think about it.
Kirk we probably learned the most about in the original series (he has siblings, some snippets of his early career, etc.), but even then a lot is left to the imagination.
So rather than reinventing the wheel, you can leave the wheel alone and tackle the fertile ground.
A really good example of the fertile ground from the original series is Amok Time. In this episode, we learn Spock has a wife, about Ponn Farr, and we get a glimpse into Vulcan culture. Yeah, there is a LOT of leftover untapped potential, even in TOS.
A thought might be to follow Scotty's early career, and his path to the Enterprise. Other characters could be addressed as well. Really, you could follow multiple backstories simultaneously (not too many though, otherwise it just gets disjointed) and end these arcs with their assignment to the Enterprise.
Lots of conflicts happened before TOS, and I'm sure several characters had interesting careers that 'earned' them a spot on the Enterprise.
In fact, Axanar jumped right into this fertile ground, and I applaud them for it! Amazing what a couple of quick lines about Captain Garth's backstory as mentioned in TOS can be developed into...
Several fan made series continue to explore the TOS setting (as well as a couple of others), which is even more proof of the untapped potential.
I suppose we could boycott the current ST franchise, but we saw how that worked out with Battlestar Galactica. Fortunately, the writers for the reimagined BG actually did a pretty good job going their own route, and I continue to love how they (mostly) didn't ignore Newton when mapping out the CG space stuff. BG made flip & burn look awesome! (So did Babylon 5 - I have fond memories of the Star Furies in action).
Sure the sweeping turns we see in Star Trek look cool, but if I was going to make one change to the franchise going forward, that'd be putting Newton back in the driver's seat. We've had at least two major franchises now that have kept Newtonian movement at the heart of their space battles (Babylon 5 being the second).
But I digress. At the very least, if enough fans could beat Paramount over the head with a petition saying 'we want the old Trek back - new actors are fine but stay true to the original please!' maaaaybe the bean counters might take notice. Myself, I'm hopeful that Axanar gets a full release, does well, and THEN the powers that be take notice...