I'm kinda surprised we can't do this by simply insulating the center-most half of the pot's bottom. When heated on a burner, this would make the outsides hotter than the core, which ought to cause our soup to rise along the sides of the pot and then roll back to the center, forming a nice current that auto-stirs.

I suspect the reason we don't do this is that eventually, even insulated, the center would still reach equilibrium & we'd lose our circulation. In order to keep this going, we'd need to change the burners, not the pots. We'd need to design burners to be hot on the outside but cooled at the center.