Mixing Metaphors-- Starfleet has a Scout Service
As a former Traveller player, the solution is simple. The Enterprise isn't a naval ship, but a Scout starship. The Scouts weren't strictly a military force, but would certainly serve in a war as reconnaissance and intel/covert ops types, also as explorers and diplomats which makes loads more sense considering what the Enterprise actually did. Your standard navy ship doesn't have science officers. Research ships have civilian/NOAA scientists detailed to the ship. The navy crews the ship and maintains the gear. Something that bugs me if they were all out to explore uncharted territories and heretofore isolated species, why they didn't have more of a science staff?!?
It'd take a lot more set-up detailing the various branches of Starfleet-- the "real" Starfleet charged with internal peacekeeping and power projection, the Scouts, SSLC (Starfleet Support & Logistics Command) (folks running repair ships, tugs, transports, resupply ships/bases or, as we used to call 'em shore whores), Marines, etc.
BTW, I'm acknowledging that a series bible in the 60's was next to impossible to stick to, AND Roddenberry seemed more into a broad strokes, build a sandbox and see what the stable of writers could do with the concept.
Again, I'm pining for the series bible that would've made that all clear from the git-go, but hey, it would've made sense to sort out why these folks were on the Big E, whether it's a cool career-making assignment or Siberia for screw-ups and misfits counting stars and planets instead of trees.
As a military nerd, it would've been interesting to see the raillery back and forth as to who's considered to have a real job vs. who's a bunch of sideliners riding the pine until things get serious.
Scouts could get bragging rights as to being loners out in the void, taking all the risks, mapping uncharted systems and planets, contacting new races, and if we go Rum Corps, getting obscenely rich from the initial trade agreements/patents/discoveries and making sure things don't get serious with newly discovered races. Your average Fleet line officer seems a bit dull in comparison.
Of course, for the Siberia aspect, exploring why everyone's sent to go count stars and make nice with the savages would be an interesting plot driver. Is it political, personal, just the normal scheme of things in the Federation?
As to the holodeck, blech, nyet, hell no. The 60's had plenty of psychedelic ideas floating around about total-immersion consensual illusions, but between nervousness about the drug culture and how to get there w/o drugs or ESP would've blown the network's mind.
I'm indifferent about the transporters. I think that they were a McGuffin that allowed an expedient personnel shuffling system. You could have stock shots of the shuttle launching from the Big-E and landing, but the details of the episode would make that a chore, you know, the coming in hot cliche, reflecting battle/storm damage, etc.
Space:1999 made that a regular feature though, (considering how many Eagles got vaporized or trashed each episode, you wondered how any of them were left or if they had an Eagle production line kicking them out like pop tarts between eps) and they had a British SFX budget, but the TOS powers-that-be decided, use the McGuffin. Beam me up Scotty it is!
Just a few random thoughts.