So all the glass does is frustrate dendrification, and I have no idea how neutral sodium/lithium can sit next to neutral sulfur and not try to give it an electron.

In an ordinary sodium-sulfur battery, you get 2V and Na2S4 at the cathode.

In an ordinary lithium-sulfur battery, you get 1.7V to 2.4V and Li2Sx (x={1,2,3,4,6,8}) at the cathode.

So this battery has something extra in it that causes the lithium/sodium to plate (without dendrification) at the cathode instead of doing some sweaty redox with the sulfur that is right over there, practically begging for that outer electron.

The carbon is obviously in the cathode for electrical conductivity. Sulfur sucks at conducting electrons, even as a polymer.

...What if it is also acting as a graphene/fullerene "shell" over the sulfur, to keep the sulfides from forming and migrating? If you let the cell drain too far, the swelling sulfides physically overwhelm the nanostructure, but if you recharge it before then, it stays in place.