The added protective earth wire will not bring you more protection unless the unit is connected to a grounded socket. Sometimes it can actually make things worse, i.e if you have a grounded extension cord fitted to a non-grounded socket. When one of the grounded units in that extension gets a fault, the live mains may be conducted to casings of all the other grounded units via the protective earth.
In the old times (but not all the way to the valve radios), the equipment usually had a wooden case and the mains was limited to the primary of the transformer and the power switch. These are quite safe without PE as long as all insulation is intact and all wires stay connected (especially the mains switch area is a bit scary in some models, I always add some shrink tube and tie wraps when I'm in there to keep the wires safely at hold even if the mains switch disintegrates).
The electrical regulations vary wildly from country to country, and did even more before EU harmonization efforts. Still in the 70s, here in Finland the regulations only mandated earthed mains sockets in rooms that had some kind of an exposed earth, like kitchen sinks and water taps. In those days, equipment that wasn't safe to use in those environments had a "class 0" plug that would not fit into an earthed socket (until some experimental person used a knife on it...). Those that were double insulated, i.e externally safe even during a fault, have a flat plug that fits into both kinds of sockets, and most modern AV equipment falls into this category.