Burl,
Your are correct, the house battery powers the solenoid. So if it was stone dead, the EM switch would not function. The engine jump start is the real reason for the switch. But I had to use it to jump start the genset and that worked too. The house battery had enough power to energize the solenoid.
Re charging, it's all about amps. The voltage has to be high enough pressure, but the amps is the volume. A squirt gun can be high pressure, but not have any volume.
I think your system is working the way it is supposed to, at least in this universe. :) :) :)