Try unplugging the remote switch wiring at the generator. Then see if the random shutdowns still happen. If they don't, I'd look for a ground short in the remote wiring that may also lead you to the failure to start the generator remotely. Somewhere that the wiring passes over or through a metal grounding surface and has frayed would be what you're looking for.
Dutch
2001 GBM Landau 34' Class A
F53 chassis, Triton V10, TST TPMS
Bigfoot Automatic Leveling System
2011 Toyota RAV4 4WD/Remco pump
ReadyBrute Elite tow bar/Blue Ox baseplate