Why does the Earth's electric field affect things so little?
The reason you don't get a powerful current is that the conductivity of air is so low. Since the conductivity of the copper wire is so much higher, the electric field itself is modified.

Basically, charges do move around in the copper wire, which means that you have to consider the entire wire to be at the same potential. This causes the nominally vertical field lines of the Earth to be warped in the vicinity of the wire &mdash; or any other conductive object, such as a human body. In a short period of time, the charge distribution and the modified field reach a new equilibrium state.