interesting thread here but lots of misconceptions
magnets work because unmatched electron spins within the material are all lined up in the same direction. this can be stable if unperturbed, but any attempt to extract "work" or "useful energy" out of them results in those spins becoming less aligned, and the magnet eventually wearing out. in a sense, this is like the magnet acting as a battery.
similarly, with atoms and electrons. electrons orbit their nuclei, and do have momentum, but that momentum is inherent to the specific type of atom, and the current state of the electron. quantum mechanics says that electrons can only occupy certain quantized momentum "states", meaning the momentum can only have discrete values in the atom. by interacting with individual photons, electrons can interact with their environment. for example, an electron in a high-energy state can fall to a lower state, and release energy, but would have had to be excited in the first place (which required energy). if an electron is not excited, it can't release any more energy, even though it does have (angular) momentum. also, to release energy, an unoccupied, lower-energy state must be available for the electron to "fall" into. for this reason, electrons in the lowest, or "ground" state can't release any energy!
You can bombard atoms with masses and the electrons will absorb part of the momentum of the masses and jump up, then drop down and release the energy as a photon. So electrons interact with momentum not just with photons. Photons are maybe an easier way of interacting but they are not the only way of interacting.
stirling engines work by exploiting a temperature difference between two thermally insulated reservoirs. a great example of this is geothermal energy (see Iceland). the hot reservoir is the hot magma, which functions as a source of energy, since the heat tends to flow towards a cold reservoir. in theory this is a great energy source, but the act of extracting it eventually will cool down the core of the planet (but this would take millennia to even be measurable.)
so yes, large temperature differences are a great source of energy, but it is still much easier to create a temperature difference by burning things or fissioning nuclei than by drilling gigantic 10-mile deep holes into the ground.
Iceland is a nice example as lava gives a big temperature difference. But it is technology that can only easily be exploited in certain parts of the world. It is by no means a universally available source of energy.
Btw, arent they totally into steam in Iceland? I heared they pump water into these hot places and leverage the phase transition of the water.
Anyway, what was proposed was a stirling engine that is practical and uses only the temperature difference between a lake and air. I say the surface you need to pump through enough usable energy would render it unusable for most applications.
Furthermore this would work mostly on the energy stored in the outmost layer of the earth. It's nothing compared to the volume of energy sitting inside our earth.
So my guess is if we used this surface potential that pretty soon the sun will not keep up with warming the surface enough to make a difference.