You say "voltage is fine". If Voltage is fine at the points then they would be working so I'm guessing its either +12v is present but your having to earth out manually, or the voltage was taken further back down the wires, in which case find the break further up the wiring? (just after clarity)

Reverse bulb earth point in likely in the boot. But its more than likely the reverse switch in the box itself at fault - easy enough to bypass to confirm that tho.

Lighter point you'll have to trace behind the dash as it likely shares a common earth point in one of the footwells. Could also be a dodgy connector, had a few of these come loose.