3rd conductor could be used to ground chassis but not required, could cause a ground loop, and I would be reluctant to do it without knowing the design. Chassis grounding is required by various safety rules unless system meets certain safety requirements, often referred to as "double insulated". apparently your rig is "double insulated", as are many other appliances with only 2 conductors.
The 3rd conductor is for safety rather than sound (generally).
A heavier conductor (for the other 2) will likely help. that is why we upgrade.
Having an unused 3rd wire will not hurt anything.
Jerry