The real fun part is what happens if/when the ground opens. Let's say you didn't screw down the ground properly in the circuit box and you had a 120v motor in the appliance connecting to one hot leg and the ground (which is also attached to the frame of the washing machine for safety). Now 5 years later the ground opens. Maybe thermal expansion loosened it, maybe the small amount of current being used by the motor browned the wires a bit more each day till current would not flow. This happens.
You now have a situation where the power goes from the hot leg, through the timer motor in the appliance, and dead-ends at the ground as there is no path back to neutral. Thus the whole dryer metal box is now "hot" with the current from the dryer motor trying to find a place to go to neutral. Now you touch it, while standing in that wet puddle next to the washing machine.
You have now become the path to "ground". And since 100ma at 120v is enough to kill a person you had better hope that motor can pass less than 100ma.
(Electrical safety, the fatal current (1987), Ohio State. Retrieved from https://www.asc.ohio-state.edu/physics/p616/safety/fatal_current.html)
That is the problem. If the motor went back to the breaker panel on a neutral and the neutral opened then the frame of the dryer would not be hot as it is connected to neutral and not the motor. If the ground opened up then the frame of the dryer would still not be hot as nothing is connected to it (however if a ground fault developed say in the dryer motor the frame could become hot which is why you ground it). The voltage differential is one thing, but if the frame ground is expected to carry a load and the ground wire becomes disconnected you basically have a hot stove waiting for people to touch it.
Interesting stuff.