The cathode was described as an ink, with sulfur as one component. The copper is not part of the cathode, but a means to conduct electrons to and from it. I didn't see anything about how this cathode was processed, so I assumed it had a gel-like consistency, and the individual components retain some physical mobility. The charge carriers would be in aqueous solution, and the liquid solute would be stabilized by support molecules to keep it in place. It might need to be liquid at first to make a good connection between electrolyte glass and copper conductor, then dries out. Even ink that feels dry to the touch can retain enough water to still be considered aqueous chemistry, right? A block of firm gelatin is still mostly water. Maybe the cathode is like that? Perhaps it is dried entirely and sealed against moisture. In that case, I'm not sure what the atoms in the cathode are doing.
2. The cathode is physically very thin, according to the diagrams I have seen. There's no room for very much of the lithium/sodium to enter it. If the lithium exited the electrolyte as ions, it could plate at the cathode, but I think most of the ions remain on the cathode side of the electrolyte glass, as close as they can get to it, then get forced to take back an electron on the cathode side. Once neutralized, they are immobilized in the glass.
3. You need something present to pull the electrons off the lithium/sodium. It can't send its own extra through the electrolyte that does not conduct electrons, only ions. The lithium is only mobile through the electrolyte when ionized. If the lithium were in direct contact with the copper, it could ionize there, but then what does the sulfur do?
4. The atomic/ionic radii in pm are Li+ 90, S 102, Na+ 116, Li 134, Na 154, S-- 170. If the mean aperture diameter in the glass is between 180 and 268 pm, lithium ions would be mobile through it as neutral atoms are immobile, and at that size neutral sulfur could enter it, but ionized sulfur could not. If the mean aperture diameter is between 232 and 308 pm, the same is true for sodium and sulfur. I know there are some macroscopic minerals useful in chemistry (i.e. zeolites) for their aperture size in relation to molecular dimensions, but I don't know how precisely we can tune man-made materials. It seems likely that we could still overproduce with an imprecise technique and then select the appropriate materials through binning, though.