Ya know, staring at that diagram something rather fundamental occurred to me: there's no benefit for all the TLs to move horizontally.
Think about it; at its largest diameter, the saucer is 2 decks high, tops. A stairwell will get me up or down a deck faster. And without lots of hallways blocked by TLs, I can also walk from one side to the other pretty quickly.
All the saucer needs is a few vertical-only TLs closer to the core, where the deck numbers are the greatest. As the decks get smaller, the need for TLs is reduced, so they would stagger as they do in a tall building. You would only need a single horizontal TL to zip down to the engineering hull.
After I had this realization, I dug out my FJ plans, curious what his solution was. Turns out, he came to pretty much the same conclusion. He does have a few horizontal tubes in the saucer, but they're relegated to runs to the EH, and a few key locations like the impulse engineering room and 'battle bridge'. Otherwise, he has a number of normal elevator tubes in specific locations to service vertical movement within the saucer.
You're absolutely right.
Unless the ship is populated by cripples, on a "day to day basis" you'd actually the crew to walk most of the time, too. Otherwise, you'd end up with obese engineers and commanders...
The main role of horizontal shafts is to provide "crossovers" between vertical shafts, and to provide "looping" to allow for reasonable traffic flow.
The ship really OUGHT to have some stairwells in various places. We never saw those in TOS. We saw LADDERWAYS, but no stairwells. Well, the answer to that is that ladderways can be cut off by hatches (much as seen in TWOK) but stairwells need to be in isolated rooms to provide "sealing" between decks (in case of hull rupture, for instance). So the reason we never saw them in TOS was because they were behind closed doors!
The other issue with lots of horizontal runs is that you create these areas of the deck which are inaccessible from other areas of that same deck. But there's an easy solution to that.
Assume you need a complete "loop." Simply stagger that loop, so that (for instance) the forward and aft quadrants of the circle are on Deck 6, and the port and starboard quadrants are on Deck 7, with four vertical shafts (that's assuming that the 45-degree placement of vertical shafts is used, as opposed to the 90-degree placement which is more implicit in the TOS design, of course).
You need very few horizontal segments. JUST ENOUGH to allow for "no roadblock" transit between the different vertical sections, and the ability to reroute and provide access to the key decks if a section of shaft is disabled.
The rest of the time, the crew NEEDS to walk!