On the "Alternative Factor" issues, we do see the crystals in a location that does not correspond to the other well-known location - the articulation frame under the floor of (one of the) Main Engineering control room(s). We see a blueshirt specialist handling them. We see Kirk's ship unaffected by the repeated thefts of crystals from the energizer, whereas removing a crystal from the articulation frame appears to deprive the ship of main power.
In "Mudd's Women", the ship came equipped with six crystals, after gods know how many had been lost to the Galactic Barrier in the pilot episode. Having four crystals at a time in recuperation would keep the ship operational - so Kirk would be pretty annoyed at the repeated thefts of recently reamplified ones, but not concerned about the survival of his ship yet.
My take on this still is that dilithium needs to be in an "energized" state in order to turn antimatter into warp power, and that "energizing" and "reamplifying" may be related to the "recrystallizing" that later allows drained crystals to be nurtured back to operation. However, the energies inserted into the crystal in this manner are not the ones released for applications. Rather, the energized crystal just channels other energies - antimatter aboard the starship, but inter-universe energies (which also have something to do with antimatter, according to the dialogue) in case of the technology of the two Lazari.