Hi Alex,
You may remember my take on this from the past, but I'll toss it out there again.
The Enterprise is one of a large number of cruiser-type starships, not merely twelve. These cruisers were set up in the "The Cage"/"Where No Man Has Gone Before" state, and carried a crew of just over two hundred, with space for as many as thirty "specialists" (in cramped quarters) for special missions, ike extra-galactic probes).
But with the advent of early "replicator" technology, much of the stowage space became available. You didn't need to carry as many spare parts, or spare uniforms, or food supplies, or so forth. You could make new water from chunks of space rock if you needed to (at a significant power expense of course). A LOT of interior volume used for cargo/payload was freed.
This new technology was fit into twelve heavy cruiser hulls. The space which was freed up was converted into additional science labs and crew quarters. Basically, they were able to fit the full complement and resources of a science vessel (less command crew, which would be redundant) into the freed up space.
These ships were the first "Federation Explorers," which eventually, by the time of TNG, is actually a ship classification (the 1701-D was an "explorer," remember)
Now, we know that the Enterprise, and most ships, operate on a three-shift system. And we know that command crew, engineering crew, medical crew, and the like during Pike's time made up something like 200 personnel, and likely this remained the case post-explorer-refit.
Finally, we know that on yellow-alert, two full shifts are on-duty, and at red-alert, all three shifts are on-duty. Some people man secondary duty stations, and some get assigned to damage-control parties, and likely some get assigned to supplement security operations. (You can't stay at Red Alert for days at a time, but you can stay at yellow alert for days at a time... this is consistent with what we see on-screen, as well).
Oh, yes, and the Captain is not assigned to a "shift." He's on-duty 24/7, though he normally would operate during first shift, I think, and normally would sleep during third shift.
So... for a crew of 430, that's 429 crew to be divided into three shifts, or 143 personnel per shift.
On Pikes' ship, he says he's "responsible for 200 lives." Let's say that he's omitting himself, If the total complement is 202, this means that the captain plus three shifts of 67 personnel each.
Sixty-seven personnel per shift should be more than enough to run all the primary duty stations for a ship of this size and capabilities... after all, on normal status, the bridge will have what, six people? Main engineering will have four?
So... engineering and operations for the TOS 1701 should make up 67 per shift.
Now, the "Pike's Enterprise" may have had a very small sciences team and a very small medical team. We see no indication of any dedicated security team, either. Maybe eight of those personnel per-shift would be dedicated medical, security, and science types? Likely, there would be a pair of on-duty security personnel (think "sheriff and deputy"... and the Enterprise is a VERY small town!), a pair of medical personnel (a doctor and a nurse... or shall we say, a "medical assistant?") per shift, and four science personnel (one on the bridge).
That still leaves us with forty-nine personnel running around the ship doing other duties... Which is quite reasonable.
Now, for the TOS ship, with the much larger crew, we still have the same number in Engineering, the same number on the bridge, etc, etc, during normal circumstances.
To be honest, I have a hard time imagining what 143 personnel would do on a day-to-day basis aboard a ship like the Enterprise. Obviously, some of them are so underutilized that the first time they're needed, they end up falling in love with a cryogenically-preserved genetic superman.
Marla McGivers is the quintessential example of how "overboard" they went when they staffed up this ship. I suspect that she split her time between "work in her cabin" (which is why she had a private cabin, unlike most of the crew) and work in the library computer complex. And as far as I can tell, she never did anything else the whole time she was aboard.
There would also be geologists, meteorologists, cultural anthropologists, and so forth... which would likely never have been found on Pike's Enterprise. During Pike's command, the science officers would have been expected to fill all such needs in a more "generic" sense. There would be a full biological research team, maybe six total (as opposed to a doctor, nurse, and lab tech on Pike's Enterprise). There would be a linguistic team... likely three or four people... supplementing the three commmunication officers from Pike's era.
Now, I'm summarizing this right now, but I went through this thought process in (excruciating) detail when I was figuring out the complement of the Enterprise a while back (I had to figure out how to quarter my crew when I did my ship layout, after all!).
VERY few of the crew get private cabins. Only senior officers and those who mainly "work from their quarters" (using it as an office), like the quartermaster, the captain's yeoman, the historian, and so forth. Maybe a junior ensign in security would briefly have a private cabin, but only until his new cabinmate arrived on-ship (likely this had been one of the redshirt ensigns who'd been killed at various points!)
Finally, it's worth mentioning that despite what Roddenberry later claimed, and which FJ was forced to conform to, the lowest-ranking personnel on the Enterprise were NOT "ensign rank." The term "crewman" is used repeatedly, and is used in a way which is consistent with contemporary naval terminology (where the word is "seaman") or air force terminology (where the term is "airman"). This refers to enlisted personnel, not officers. And "Ensign" is the most junior officer rank (well, technically you can argue that "cadet" is an officer rank, but that's a dubious argument). If you use the modern-military model, you'll get a ship with at least one officer on duty in each department at any given time, with the rest being enlisted.
So, if you assume four personnel in main engineering, typically one would be an officer and three enlisted crewmen. The doctor would be an officer but the nurse would be a crewman. And so on.
The real issue, as far as I'm concerned, is how the "science specialists" managed to keep busy and be productive, any time the ship wasn't in orbit around an interesting planet...
