Are unearthed metal cases actually unsafe?
if there is a fault with a device, current will flow from Live to Earth, hence you don't feel an electric shock.


There are two aspects to this : if there is a catastrophic fault (low impedance, allowing high current to flow) the traditional fuse will melt, and protect you by disconnecting the live circuit.

However if the fault is a partial breakdown (perhaps moisture or degraded insulation, which may get worse, but only allows a small current at the moment) the case is either at a dangerously high voltage, or safe because the current is conducted to earth. 

Now the earthed case is safer because it is at earth potential, and furthermore, if the current reaches 10-20mA, the live circuit will be disconnected when the breaker trips (RCD in Europe, GFCI in the USA).

In this case the un-earthed case is unpleasant but probably still not lethal, because when you complete the circuit well enough to conduct 20mA, the breaker will trip. And unless you have an unusually weak heart, you will probably survive. (You do test the breakers every year to make sure they aren't sticking, right?)

In practice, if you are deliberately designing an un-earthed case, you will adopt "double-insulated" design practice, in which any single failure (such as insulation breakdown in the transformer) cannot leave the case live. One easy way is to make the case itself plastic and therefore non-conductive! The safety approvals procedures and regulations are designed to ensure that someone independently verifies that these safety practices are correctly followed.