And the result... drumroll please...
It appears that I was over-taxing the USB bus power with the Balance. After moving the Balance to a powered USB hub, it works exactly as expected. In other words, when the computer is off, the Balance is off. When the computer is on, the Balance turns on.
So... for anyone having similar trouble, before anything else try connecting the Balance to a powered USB hub.
Thanks to everyone for helping out, and thanks to Propellerhead for the support.