On the issue of terminology, we know that the TOS ship had a hangar deck (sometimes mislabeled as hanger deck):
However, we could argue that this deck was the one below the flight deck or shuttlebay. After all, we never see any of the familiar flight deck/shuttlebay features in the same shot with the door signs that say "hangar bay" or "hanger bay". And if shuttles are prepared for launch on the hangar deck, and then raised to the flight deck for the actual launch, it makes perfect sense for those doorways to lead to this lower deck rather than to the upper...
The TOS-R versions of the episodes try to add some detail to the grey backgrounds of the "hangar decks" beyond the doorways, but this detail never really matches what is shown on the flight decks - especially since TOS-R adds a bit of detail to the flight decks as well, including windows on the floor level that should be quite visible through the doorway. At least before TOS-R, we could argue that the shuttle seen beyond the doorway was positioned so that it blocked the trapezoidal alcoves from being seen, since those might in theory have been the only floor-level detail "back then"!
Let's also remember that whenever we see the shuttle through the "hanger deck" doorway, it's rotated differently from what we just saw in an establishing "flight deck" scene, at least in TOS. All the more reason to assume that it was moved down to a different deck by the turntable...
For all we know, the turntable-elevator features some sort of an envelope that allows for the lower deck to be pressurized while the upper one is not. In any case, it doesn't seem as if Kirk's ship carries too many auxiliaries during TOS: at most two are ever seen at a time in TOS-R - and every time we see the full-size prop in TOS, its surroundings are distinctly devoid of other craft (doh!).
TAS of course shows the upper level or flight deck crammed full of auxiliaries - six in "Mudd's Passion". Then again, it wouldn't be impossible to see an aircraft carrier's flight deck jam-packed with fighters at times. Sometimes carriers do take aboard more planes than they can actually operate (they were very practical as "aircraft transports" back when aircraft couldn't be ferry-flown overseas), and sometimes they move all the planes up from the hangars to clutter up the flight deck, perhaps for parading or other PR reasons, perhaps because the hangars need a bit of cleaning... I don't have anything against Kirk's ship carrying six shuttles in TAS, as long as most of them would "normally" be stowed below decks.
...After all, it does seem as if launching a shuttle requires quite a lot of space. It's not like precisely floating this small spacecraft out of its berth on thrusters, with only centimeters to spare - in TOS-R, we see on several occasions that the shuttles wobble and drift when "taking off", especially during Decker's crazed flight in "Doomsday Machine" but also during the supposedly routine takeoff in "The Galileo Seven". Mudd must have been exceptionally
careful when taking off in that TAS episode...