Have they really defeated the second law of thermodynamics, and created a heat engine that does not rely on a difference in temperature, but harvests molecular kinetic energy? If so, it should also work many places above ground, and win a Nobel.

Thermal physics second principle entropy needs at least 2 heat sources at different temperatures to convert earth heat into energy, by cooling the deep earth, by any method. Thus this battery after discharging at a low room temperature recovers because as they write in their abstract : " the battery characteristics were restored after discharging by placing or burying the battery in a heat source", i.e. this battery is recharged by heating it "buried" into earth. Thus to work, this battery need to be moved, up and down, between low and hot temperatures, i.e. moved between surface and deep hot rocks below earth surface ( at least 1 Km below at around 50°C at 1°C for each 30m deep ) and this is contrary to the statement "without needing a middle-ground carrier like water or steam", because moving up and down 1km, the battery is more difficult than moving the water. This new type of battery is interesting, but we must be careful to not overestimate its possibilities. Otherwise the efficiency, not indicated, is very likely quite small with respect to Carnot second principle maximum efficiency, already small, 50/350, ratio of diifference temperature 50°C to maximum temperature.

Whatever happened to freon motors? Obviously some freons are a hazard to the ozone but there are many more chemicals that could fill the void for a low temperature heat cycle.

I agree with Bob Stuart, if proved true, "defeated the second law of thermodynamics", not only they win a Nobel, but a revolution in physics, turning all up down. It becomes possible to move on boats and motors extracting the energy of sea, leaving behing them icebergs, an nearly infinite energy !! Thus it is extremely unlikely, nearly impossible and we must search where are the basic errors !

No laws are being broken. No free energy is being created. Convert the core's heat into electricity and the core (ever so slightly) gets colder.

Was there a valid reason that Thermocouples were completely left out? Explain that to us.

If it does require alternating hot and cold, then it's not a new phenomena. I read about it as an effect in lead-acid batteries, where the change in temperature boosted the charge (converted heat energy into electrical charge separation). As a heat engine, it wasn't very practical.

There is unlimited geothermal energy, worldwide, some is easy to access, some not, but like oil wells, eventually the more difficult will become viable. Drilling horizontally into volcanoes may be one way, with the right precautions. Otherwise, hot-spots like Yellowstone, Hawaii, and Iceland are obvious locations. Batteries tend to be expensive to produce, short-lived, and fragile, relative to steam power. So this invention may seem hopeful, but is also a long way from fruition.

what about other heat sources that do get very hot then cooler, say the exhaust system of the humble ICE?

Pilot these offshore of Japan where underwater vents from volcanic activity is present, then use the heat and surrounding colder water to make electrcity. Once the juice is running, put the electrcity to work by splitting water into Hydrogen and Oxygen.
Pump the hydrogen to shore and put it to use Diffuse the oxygen in the ocean like a big fish aquarium keeping the fishes happy and healthier.