Hmm. If you measure from the battery "-" to the frame and engine, and read battery voltage, that means the battery negative is not connecting to the frame and engine. There really isn't any other explanation. A few things like ignition coils, field coil, various lights, all get power from the battery but ground to the engine or frame. So if the connection from frame to battery is no good, you'll read battery voltage from battery neg to the engine and frame through all those things.
If you connect a jumper from battery neg to the frame, it shouldn't short and melt the jumper - just power up the stuff that's on.
The negative cable to the frame is pretty large and if something is shorting out, the harness wire will melt and burn without that ground cable even warming up noticeably.