While there are plenty of toy Stirling engines, they do have real world applications. They are used as heat pumps in cryocoolers, and there are even variants with no moving parts, the Thermoacoustic Sterling Cryocooler.

I could see it used for fixed-load low-power uses; maybe a trawling motor?

Well if the failure mode under high load turns it back into mechanical it still seems useful. Not sure what the Eddie current losses mean though.