What we know, for certain, is that without the dilithium, the TOS ship "loses power."
Perhaps, perhaps not. When three dilithium crystals in a row were burned in "Mudd's Women", Spock suggested "rigging a bypass circuit" so that the one remaining crystal wouldn't be a risk factor. Scotty couldn't oblige because "the whole converter assembly" was damaged. The implication might be that the ship could run at warp or otherwise traverse interstellar distances without dilithium, if only the converter assembly (whatever that is) were in working condition and some special arrangements were made.
In "Alternative Factor", all four active crystals (nice continuity with "Mudd's Women"!) were damaged by the hiccuping universe, and were brought to some sort of a nurturing station. They were then stolen from there, two by two. At no point was the ship said to be deprived of power. Either Scotty managed to rig a bypass circuit this time, or then there were additional crystals that could be taken from storage and put in place of the ailing ones.
Just to nitpick. But it pays to be careful.
Fair enough... but I was specifically talking about conversion of the m/am reaction into power, not necessarily the "auxiliary power" (generally assumed to be the fusion powerplant output)... just the matter/antimatter system.
An interesting point about that "bypass circuit." Let's think about that for a moment.
Perhaps the m/am reactor subsystem (which I personally think of as being in the nacelles, but this works just fine if it isn't, too) produces usable power by one system - incorporating dilithium. And maybe the auxiliary power system (fusion, whether or not tied into the impulse drive system) uses a somewhat different, but not entirely incompatible, system, not using dilithium.
Well, yes, in "Mudd's women" they were crippled without the crystals. That's a pretty glaring design defect, and I'd expect that after this episode, the ship spent the next couple of months in drydock getting this design defect fixed, so that you didn't need the same "energizer" for all possible power systems.
I'm extrapolating, sure, but it seems reasonable, doesn't it?
So, after that, you have not one but TWO "energizer" subsystems on the ship. Only one (at "main engineering") requires the use of a dilithium system. The other is far less efficient, and would probably even be damaged by extended use with the m/am output, but you could "bypass" to use that as a short-term fix.
I think that "fits" reasonably well with what we've seen, doesn't it?
(FYI, in my ... currently gathering a bit of dust... 1701 interior layout, I've got three "energizer" - aka "rows of tubes" - assemblies, with three engineering-deck facilities. Two are in the primary hull, to either side of the impulse deck, and one is at the intersection of the nacelle pylons in the secondary hull. In this case, the one in the secondary hull is the one with the dilithium crystals.)