If you had a normal electrochemical cell, with a sodium anode and sulfur cathode, and liquid electrolyte, the sodium would release an electron into the circuit

If you had a normal electrochemical cell, with a sodium anode and sulfur cathode, and liquid electrolyte, the sodium would release an electron into the circuit and hop off the solid anode into the liquid electrolyte. On the other side, sulfur would grab electrons from the circuit and hop off the solid cathode into the liquid electrolyte. In the middle, the ions would keep their charges and physically arrange themselves such that the charges are balanced in every direction. This would be an energy minimum, and it would be hard to reverse. Trying would probably cause the electrolyte to decompose instead of redepositing the original reactants onto their electrodes. It would not be rechargeable. As a physical metaphor, you have two mutually-attracted boulders rolling down from opposite ridges into the same bowl-shaped energy valley. When they collide, they strike with such force they turn to sand. In the rechargeable battery, you prevent that last step by putting a giant springy foam block in the center of the valley, so that the boulders can't touch. You prevent the sodium and sulfur from coordinating physically to cancel out the physical force from separated charges. Then to recharge, you tow or push the boulders back up the hill. I think the "same metal at the cathode" might be an artifact of putting the springy foam block in to keep the boulders intact. You couldn't switch the anode and the cathode any more easily than you could swap the elevations of the ridges and the valley. That is, it takes as much energy to do the physical rearrangement as it would to charge the system electrically.