Off Topic but relevant nevertheless
"Why is the neutral wire in a 240 circuit the same gauge as both phases (hot wires). Shouldn't the neutral wire be much larger to carry double the current?"
No.
Current in an AC wire "only flows half the time"
If that current can be "filled up" then the wire will be "full".
The 2nd phase wire wills the neutral wire up with a mirror image filling the sine wave gaps.
Three wire to two wire adapter plug test...
Those little plastic adapter blocks that will allow a grounded three wire plug to connect to a two wire outlet is a great diagnostic tool.
You'll need a non contact AC voltage tester for this...
Unplug ANY metal cased appliance then plug it into the adapter then plug the adapter into a wall socket.
Wave the ac detector over the metal of the appliance. If it sounds off there is a fault inside the appliance.