By my lights, the primary hull is primary (i.e., where the crew is housed). I have hard time imagining the neck being a human-friendly environment.
There are four main shapes that make up the enterprise. The primary hull, the secondary hull and the warp engines. The whispy connecting structures appear to be just that - connective structure. If the E had no warp drive, you would not have those popsicle-stick supports. If the E were a Miranda style ship (a Bird of Prey type design in TOS terms), there would be NO neck, because there would be nothing for the neck to connect to. These three parts are instrumental - they serve the purpose of connecting our saucers and cylinders.
The neck connects the primary hull to the secondary hull. This means wires, pipes, and elevators. It's hard for me to imagine that there are not internal supports in there to brace for torsional stress, so that's some room (how much?) gone too.
It seems to me that this part of the ship is basically access tubes, maintenance closets and other odds and ends. It's almost purely instrumental in linking one part of the ship to another. The main business of the ship being conducted in the main areas.