More efficiency with inverter pumps

Solid state heat pumps of today have dismal efficiency. Exceeded only by their inefficiency in the reverse direction. I'm sure people are looking for materials with a better Peltier (electric to thermal) or Seebeck (thermal to electric) coefficient, but none are surfacing as far as I've seen. And bisnuth telluride is not exactly your cheap and plentiful materials system. Heck, the only place you see it compete is in niches where there have to be no moving parts, or the thing has to run off DC low voltage (like those lame 12V car coolers that can't even make ice). Even there, a cube fridge ($58) and a small inverter ($50) would cost less and do more. There's not even any mass-market full-size refrigerators using solid state cooling, and those have a tiny heat throw relatively speaking. I've played with thermoelectric modules, some. They are niche-only technology.

Inverters can exceed 90% efficiency routinely. This is not the big loss term, in context. It's probably better than the raw charge/discharge cycle efficiency of any battery, certainly better than any compression refrigeration system's thermodynamic efficiency. And the solar panel?

The problem is hardly "analog" versus "digital". And "analog" vs "solid state" is really not even a valid distinction. The problem is inappropriate use of a "highly refined" energy source (electric, with umpteen conversion losses in the chain) to do a low down dirty thermal job. Heat to fluid to mechanical to electric to mechanical fluid to heat is just not as slick as a heat-heat machine (with, no doubt, some supporting electric pumps, but these should be as little energy input as possible).