Why earth ground connection does not leak current?
In reality there is current in the ground connection, but not enough to measure above ambient noise level. 

Below is your equivalent circuit.







simulate this circuit &ndash; Schematic created using CircuitLab

ADDITION

As such we normally ignore that leakage current in DC circuits. So let us look at your question another way.

For the moment consider what would happen if an electron fell down that "drain" to ground. What happens to the rest of the circuit?

Suddenly it will have a missing electron and will have a net positive charge. 
The ground now has an extra electron and has net negative charge. That means there is a reverse voltage on the ground wire which will immediately cause the electron to return to the circuit. 

The opposite would happen if somehow an extra electron were to arrive from the ground. The circuit would be at a negative potential and ground would be positive. The visiting electron would be immediately repelled.

In actuality, that returning force is what prevents the electrons from crossing the connection in the first place. It is a self stabilizing state.