In airplane design, observation galleries were typically shunted to places where they did not interfere with more serious things such as propulsion or steering; sometimes these were built into the wings, even. The neck might be the only part of the ship deemed worthless enough to receive a crewmen's lounge or somesuch.
However, it should be taken into account that a lot of space and attention is given to recreation facilities on the TOS hero starship. There are multi-deck recreation rooms ("Let That Be..", perhaps the same facility as in ST:TMP?), and numerous smaller rooms; a bowling alley was once mentioned, even, although probably only in jest. Significant effort might be given to fitting an observation deck or other stargazing facility aboard the ship as well, then.
The best counterindication to that might be that Kirk didn't take Lenore Karidian to such a facility! Instead, we appear to witness an observation gallery next to the shuttlebay, as if that were the best Kirk could hope to offer. Certainly the kinked gallery there, with the angled walls, would fit better anywhere else but
If the rows of windows in the neck don't provide a spectacular view for visiting babes, then, we might have to speculate that they are off limits to civilians, even when attended by the Captain himself. That might mean dangerous and unergonomic spaces, or spaces with military secrets, or spaces dedicated to scientific solitude and calm. Why any of these spaces would require windows is very difficult to imagine, however.
My best guess currently is that the windows there cater for the discerning passenger. That is, the neck is an otherwise useless part of the ship and thus accommodates numerous passenger cabins. Some may be permanently configured for nonhuman passengers with special environmental requirements (this is a popular idea in older Trek literature and speculation, too), and many may await use empty for extended periods of time. However, Kirk once had to move down there as repairs on the upper decks of the ship were still ongoing following (or preceding) the shift to a lower-profile superstructure...