The transporter unit ionizer, which "negative" Kirk shot with the phaser after being neck-pinched, did indeed create problems, but the transporter as a whole was already down due to the previous malfunction. Before that, all work relating to the transporter seemed to have been done in the transporter room. Granted that there is no reason why additional work couldn't have been going on elsewhere, but we never saw it in this episode. The only other time I can think of off the top of my head where transporter-related repair work was shown being conducted outside of the room itself was Scott's work on the main transporter circuitry in a Jeffries Tube in "The Doomsday Machine" - which seemed to have been located near the room itself. I always figured that the transporter ionizer in the engine room, despite the technobabble-sounding name, was simply a conduit to get power to the transporter room - which I feel is supported by Spock and Scotty's being able to bypass the circuitry and connect the transporter directly to the impulse engines.
I think that "shuttlecraft" is also the word when describing them in plural - I believe there is a reference in "The Omega Glory" to the effect that "all four shuttlecraft" were still aboard the Exeter
. Not sure about the turbolift, but I always figured the word "turbolift" was used in the singular to refer to the system as a whole, not individual cars.
It's probably overall another case of over-analyzing a TV show that never really was intended or expected to come under the level of scrutiny that it's received over the years. All I'm saying is that the episodes of Star Trek TOS in general, and "The Enemy Within" in particular, do not support the idea of the Enterprise
having multiple transporter units, although it doesn't make sense for there not to be. Of course, "The Enemy Within" doesn't support the idea of the Enterprise
having shuttlecraft, either.