Didn't Redjack put the lift in free-fall? That seems to support the idea that he shaft has gravity or rather, that the entire ship has a gravity field that works as if the ship were a skyscraper on earth. So while it makes sense for the shaft not to have greavity, it behooves us to remember the original creators had neither the time nor inclination to think things through as obsessively as we do.
Of course, that every deck has gravity plating, rather contradicts that. If the skyscraper analogy were correct, only the lowest deck would have gravity plating that work for the entire ship. The thing is, that Redjac could indeed make a lift go in "free-fall' without there being a skyscraper gravity. It turns off the propulsion, and the lift will simply keep going in the direction it was going - essentially free-falling - until it slams into something. Give it a little push with its propulsion if it were standing still at the moment of turning things off.
Unless the deck plating just "evens out" the gravity from the main gravity generators. So that you don't get 2G near the generator or .5G on the bridge. In a small scale, they can give their own gravity for things, but on a large scale it is better to use them in conjugation with other equipment? Just a guess, not canon.
Well, that could apply to the basic "deck plating," sure. You'd be defining "deck plating" as a form of wave-guide that would channel gravity and cause it to flow as desired. Sort of like how the deflector "grid" on the hull isn't what creates the shielding (that's the "shield generators") but it IS what causes the shielding to be distributed evenly over the hull.
However, there's on-screen evidence that's worth considering regarding the lifts. We know how big the ship is, and we see, several times, the lifts moving from inside, yet the passengers aren't being jarred and jolted when the lift starts, stops, changes directions, etc, etc. The explanation, in-universe, is that the lift cars each have their own internal gravitation and acceleration-compensation fields.
If this weren't the case, the people in the cars would have to lean from side to side when the car shifted direction, have to "lift up" or "scrunch down" or whatever... they'd have to react to acceleration forces. They don't... so there is none as far as they're concerned, inside... right? You've been in elevators which move MUCH more slowly... or been in airport shuttle-buses where you'd have to stand. You know what I'm talking about, don't you?
SO... if there was gravity in the shafts... you'd then need to have antigravity on each car to move the car... and then another internal gravity system inside. THREE systems present, with two serving no purpose but to cancel each other out, and only one serving any useful purpose.
We all know why the stuff was presented as it was in TNG and in ST-V. It was being written about by people who were thoroughly grounded in "earthborne" mindsets... and HERE, elevator shafts are dangerous things you can fall down. They just didn't think things through on a practical level for a spaceship.
FYI... another thing I'd do if I were going back to 1964/65... I'd make the little ladderways also be "zero-g." You wouldn't have to do anything with them very often, but just a few "wire" sequences in those little zero-G ladder alcoves would've sold the "they're in space" idea very effectively, I think.