an improvement over the design of older ships that landed in their entirety like the USS Horizon
Hmh? The ship of that name in TOS probably had transporters, as the Iotians were familiar with the limitations of that technology. All nicely in line with ENT that shows transporters in regular use from the get-go...
pretended there had always and only been dilithium from the start
Actually, the effort was made to develop a family of lithium products: dilithium, trilithium, paralithium, the unnamed type that made "Threshold" possible... It would be rather natural to assume that "lithium crystal" is just the catchall name for all these types of lithium-containing crystal. For all we know, Kirk's frontier starship was of a "multifuel" or "flexifuel" type, accepting the inferior paralithium in a cinch.
As for "lithium cracking", this need not refer to lithium being cracked (a process never really called by such a name in reality, as "cracking" is already reserved for a specific petrochemistry process). It might just as well refer to cracking taking place with the help of lithium. Remember that the place was visited by ore transports - and it would make very little sense to haul ore to this remote location from more central ones, but a lot to haul it to that location for refining and the eventual hauling of the end product to the distant civilization. Why haul in lithium ore for cracking, when lithium is actually fairly abundant everywhere (as long as we accept the ubiquity of Earth-like worlds)?
Obviously, the transporter system works fine just with battery power (as it did in Star Trek II to beam down to the Genesis Cave after Khan had crippled the Enterprise)
To be sure, the ship's transporters only moved our trio of heroes to the Regula I station, across a pitiful few hundred meters. The station's transporters accomplished the further feat of reaching the Genesis Cave. But that already tells us that transporting without the power of a starship's warp core is not only possible in emergencies, but supposed to be regularly achieved by the station! It just leaves open the exact nature of the station's powerplant.
But thanks to the dilithium crystals enabling warp power, the warping of space probably enabled resupply of fresh matter (especially in a solar system) for both the nuclear fusion reactors and the matter-antimatter reactors of the Enterprise.
Also, if impulse drive combines power from fusion and propellant from fusion exhausts, it is likely to consume onboard resources faster than warp drive - a curious fact indicated in "The Doomsday Machine" where sustained impulse maneuvering is a pressing problem for fuel reserves whereas warp maneuvering never ever was. Impulse movement might involve shooting out both waste helium and
unfused deuterium, as the latter would be not just an available energy source but also the one available source for propellant mass (unless the ship carries a separate substance for propellant, which would be a bit silly - or unless impulse drive involves no Newtonian propellant exhaustion, which actually sounds rather likely).
FARRELL: Preorbital course locked in, sir.
KIRK: That's the last time I'm giving an order twice, gentlemen. We're down to battery power, and we're low on that.
SCOTT: It'll get us to Rigel 12, sir, but it'll be a shaky orbit.
can be interpreted in yet another way if need be:
Might be that the ship is moving on impulse power, or even on warp power (because the loss of the final crystal was never confirmed), but battery power is being used for all non-propulsive applications, including sensor sweeps / computations such as "locking preorbital courses". Without enough oomph for the sensors and computers, the course will be poorly calculated and will give a "shaky" orbit.
This is, after all, the only time batteries might
be directly associated with propulsion in Star Trek...
SPOCK: Except for secondary systems, everything is out, sir. We're on impulse power only.
SPOCK: If Mister Scott is still with us, auxiliaries should be on momentarily.
Might be that the tapping of auxiliary power out of the impulse drive is an option, not an automatic thing. The engines can simply be used to move the ship at sublight, without switching on the power taps. Of course, the taps may be damaged, in which case it takes some effort to get the "auxiliaries" "on" even when the engines are running smoothly.
As for the deeper meaning of "energizer", I'd go for the invigoration aspect. In "The Alternative Factor", the energizers are needed to restore dilithium crystals to working order after they are drained. This sounds like a very transient thing, a rare, emergency "vitamin shot" involving a special piece of machinery. After the "vitamin shot" is over, the crystals apparently are to be taken out of the machinery and will need no further energizing; they have returned to their normal state of not being drained.
Doesn't sound that they would be the main energy source (or the main energy storage medium) of the ship, be they energized or not. Energy is still available, and is being pumped into the crystals from the energizer - clearly, it's not a case of the ship having lost its fundamental stores of energy. But properly energized dilithium crystals appear vital for getting the energy from its mysterious stores into the various applications, here and in ST2:TWoK. The procedure in "The Alternative Factor" shows that the energies of "energizing" are just a fraction of the energies actually in storage. The situation appears analogous to that of transistors, then: small energies control large energies. And having dilithium as the "semiconductor gate" of a transistor-analogue is not all that different from what dilithium does in TNG and ENT.
Naturally, it would be nice to have "online energizing" where you don't need to yank out the crystals if they happen to get drained. A "mains energizer" might be a device to meet that need, but your warp power will be locked out of reach if this device fails to keep the crystals in proper energized state for their role in channeling that power...