[Arran]

Well....Zenith Companion, if you are not careful, will kill you. It uses the outlet directly, and the chassis is basically connected to the outlet. Directly. They rectify the line directly, no transformer or anything.

Good thing - a GFCI outlet will trip.

What you have found is pretty typical of an All American Five series string transformerless or AC/DC set, they apparently did the same thing in Britain which is even more scarry given the 220 volt line voltage. The Canadian Electrical code forbid that power supply design for radios until the manufacturers made a few changes like installing a floating common negative and semi isolating the chassis from a direct connection to the power line with a capacitor and a resistor. In Canada all of the manufactures did this, in the U.S some companies did and some used the very hot chassis design of your Zenith. I have one American AC/DC set where I am installing a floating common negative, I had to rewire the set anyhow thanks to some previous owner's meddling, and I am concidering doing the same with the Fada model 1001 that also has a hot chassis.
Regards
Arran