I don't think it's that simple. For one thing, while the dimensions of the two ships are comparable, the shape of the starship means there is far less actual volume, with even less of that being usable as standing space. For example, the saucer is over 100m wide, but only one deck actually spans 100m, the rest of the decks in the saucer are half that size or less, with volume decreasing by 8x for every 2x reduction in width.
1/3rd of the starships's length is all engine. The neck is only about as wide as an aircraft fuselage, the entire back half of the engineering hull is made up of hangar bay and pylons and the hangar deck itself is too small IMO.
Comparing the aircraft carrier, the island alone is about equivalent to the central decks of the saucer or the forward part of the seco hull. The flight deck and hangar decks are huge, but they're also manned, habitable areas.
And the other big problem is the one I mentioned earlier about density. Do a Google image search for "nimitz class CIC" or "Nimitz class bridge" and you'll see images of extremely dense, cramped rooms compared to the spacious, sparsely populated sets we saw in TOS, and the corridors on a carrier are at most 1/3rd the width of the TOS sets. Also, they use ladders, not turbolifts.
And then there's the question of crew quarters. As far as I remember, the only time we ever saw bunks on the Connie was in The Undiscovered Country, and even those dorm rooms were downright luxurious compared to what most officers (let alone enlisted staff) could expect on a modern naval ship. From seeing Uhura's quarters, we can infer that everyone down to at least Lieutentant had their own cabin on the Enterprise.
