the triple deck high shaft (in the cargo bay) that Kirk enters in from is not present
Hmm... There is a triple deck shaft in the secondary hull all right, somewhat aft of the bright dot that indicates Kirk's current position. Whether that's too far aft or not can be debated.
But yes, the big issue is that Kirk doesn't ride up or down that shaft, but horizontally. I mean, he has
to find a horizontal ride eventually, in order to reach the primary hull. But there are limited possibilities for this on the upper "balcony" deck where he was standing, as we can clearly see the vertical shaft doesn't branch off horizontally there.
Then again, the deck where the horizontal shaft is displayed to exist is also the only one where it can plausibly exist, this being the widest deck in the secondary hull and the only one with lateral spaces for such a thing. It would perhaps be more satisfactory for Kirk to first take a ride up or down and then go forward along a shaft atop the cargo hold (but there's Engineering to block that ride, and indeed the horizontal line up there in the diagram appears interrupted), or below the floor of the hold (where there indeed exists another possible route in the diagram).
I guess Kirk would first have performed a complex maneuver where the lift goes down three decks, turns to port away from the centerline, goes back up two decks, and reaches this lateral horizontal shaft... But that's not as elegant as simply going down three decks and using an underfloor shaft, plus
the top diagram only shows centerline lengthwise shafts, not lateral ones.
there is a horizontal shaft that would stick right out into the recreation deck.
That'd depend a bit on where the deck is supposed to be. The twin vertical shafts would rather naturally place it right under the bridge, where there's plenty of room for it and where a three-deck recreation facility already existed back in "Let That Be". The pair of horizontal shafts would then go aft beneath the rec facility for a while, and jump up one deck after clearing the aft wall (the one with the display screens that imitate portholes and occasionally show a view of the ship's stern).