I’ve often wondered if late if rather than a “wet wall”, if houses should have a “hot wall”. Could I run a thermal loop past the fridge and a couple other appliances and either shunt it outside in the summer or distribute it around the house.

Total BTU aren't that great. Would (somewhat) work for a very small apartment, for any big house you would have a better experience making a proper thermal insulation.

Given that people are pointing out that heat recovery from washers doesn’t really work because they alternate between filling and draining (so there is no counterflow except incidental from other house activities), I’m not so sure that would be true.

Particularly if you use heat pumps instead of basic plumbing to achieve the transfer. Dump heat into a glycol line with a reservoir.