Why does working processors harder use more electrical power?
The rate at which these gates are turned on and off seems to have no relation to the power used.


This is where you are wrong. Basically, each gate is a capacitor with an incredibly tiny capacitance. Switching it on and off by "connecting" and "disconnecting" the voltage moves an incredibly tiny electrical charge into or out of the gate - that's what makes it act differently.

And a moving electrical charge is a current, which uses power. All those tiny currents from billions of gates being switched billions of times per second add up quite a bit.