Why doesn't current flow through the neutral to earth?

The 'hot' wire delivers current from the power station into the load, which could be any electrical appliance such as, for example, a fridge, a light or a motor.

The 'neutral' wire is there for the specific purpose of carrying the current coming out of the load back to the power station. If the neutral wire was not there, there would not be a complete electrical circuit from the source (power station) to the load and back to the source.

In normal operating conditions the connection from the neutral wire to the Earth does not carry any current.

There should always be 0 V (zero volts) between the neutral wire and the Earth. i.e. no potential difference should ever exist between the neutral wire and the Earth.

The "neutral to earth" connecting wire is fitted to ensure that no part the neutral side of the circuit can develop a high voltage above Earth. If that connecting wire wasn't in place and the circuit had no other protection device (such as a GFCI or RCD) fitted, a very dangerous electrical shock hazard could be present which could electrocute the people using the appliance.

That is why any grounding wires, fitted either to the appliance or to the breaker box, should never ever be cut or removed.