FWIW, the TNG manual in turn calls these things "sequence initiators", and says that when the operations console starts the energizing sequence, "this process can be controlled manually at operator's discretion". Essentially, Sternbach and Okuda appear to be saying that the part where the sliders come in is when O'Brien turns people into phased matter that goes into the buffer tank, and this can be done quickly or slowly, by controlling the energizing of the phase transition coils via those three sliders.
(Once the victims are in the phased matter form, in the buffer tank, the system shoots the stream to the target at a simple keypress, or on automatic, about four seconds later.)
Basically, then, the sliders fire up the machinery, and offer a chance to abort if things don't start out too well. Which suggests that this phasing thing only sort of tickles at first, and doesn't pull the victim apart; his or her internal forces probably snap him or her back to shape if O'Brien lets go of the sliders in the early stages. It's only after some tickling that O'Brien has to commit, slamming the sliders up all the way (in TNG, they apparently always go up for "go", as that's how the Okudagram is set up).
Whether we can retroactively apply this to TOS... Why not? It explains absolutely nothing about why there should be exactly three sliders, but it does explain the sliding act itself.
As for the triplication thing, perhaps it's simply a safety measure? As in, you can accidentally manipulate any single control, no matter how foolproof, but not two, and certainly not three that require careful use of your entire hand (and while it's better to be safe than sorry, four would be overkill). In that case, though, one slider should not do the trick under any circumstances.
Timo Saloniemi