Returning to the issue of the pylon for a moment, is there anything to say that those (or indeed any of the) round and rectangular "windows" seen on the ship have to actually be windows
Throughout the three years of TOS we only ever saw 3 confirmed windows - two in Conscience of the King
(which were square) and one in Mark of Gideon
(which was more rectangular, but still different from the ones on the Enterprise model).
Additionally, MOG showed us that portholes have both interior and exterior shutters (this feature nicely deals with the discreprencies in interior and exterior window shapes)
So, could some or indeed all of the "windows" seen on the flyby shots each week have in fact been sensors or something? It would certainly explain why there are so many of them in the engineering hull! I mean, shouldn't those engineers be working instead of stargazing?
It's enough to compare the TOS Enterprise with the Galaxy class starship a hundred years later; we see the Galaxy has a vast and extensive lateral sensor array on the rim of the saucer AND the engineering hull, in addition to a ring of sensors on the dorsal and ventral side of the saucer. The TOS vessel has only those glowing domes on the saucer and one on the back of the secondary hull; those would seem to be inadequate for a ship on a mission of exploration IMO.
The dark "windows" probably contain optics/telescope elements intended to image distant objects across the electromagnetic spectrum, from radio wave through X-ray and everything in between. The lit windows are probably active scanners, some type of radars/lidars, spectrometers, and a few might even contain ordinary searchlights.
I might even go so far as to guess that the observation lounge from "Mark of Gideon" might have been a sensor bay that was intentionally left empty so it could function as an observation port. The sensor elements of the Enterprise would all be housed in those sensor bays so they could be serviced by engineers in a shirtsleeve environment.
They could be similarly 'transparent to radiation' the same way the domes were labeled - you can put different sensor devices behind them, maybe they're on a turntable so 5 or 10 devices could all be aimed out of a single unit one at a time.
Or, hey, maybe some are widows. Maybe some are sensors. Maybe some are the self illumination lights for the hull. Maybe some are RCS. Because we don't see visible RCS or weapons in TOS, so I wouldn't expect that we can see all the sensors on the TOS E...I don't think the domes are the only thing.
There's also two 'windows' in the engine pylons...I hope those aren't windows, but they could be almost all of the above things (RCS, sensors, lights...)
I think to get an answer about the neck, you need to make some interpretations on all the above details.
"If you need a holodeck to make an interstellar starship on the bleeding edge of the unknown
interesting, something is seriously amiss."
- Straczynski & Zabel