We do indeed generate electricity, though it is a tiny amount. The electrical energy available from neurons firing is somewhere around 85mW. There's plenty of power going around through the brain. The brain consumes 20W of power in the form of sugars, but most of it does not go to the electrical impulses.
For some points of reference, a USB device is allowed to draw up to 2500mW (2.5W), and that can go up to 10W with some negotiations. A hairdryer may consume 1000W and the Nisan Leaf comes with a 80,000W engine!
We generate electricity by ion pumps. These are small molecular structures on the surface of neurons which use energy (in the form of ATP) to create an electrical potential. If the neuron fires, this potential changes rapidly, propagating a tiny electric charge.
The power of the body is not in its electrical subsystem. The power of the body is in its ability to carry fuel (glucose) and oxidizer (oxygen) to all the cells in the body. If the body wants to transmit power, it uses these tools rather than electricity. Your muscles are stimulated by electrical impulses, but the brute lifting is done purely chemically.
We have looked at harnessing that bioelectric energy. There have been efforts to make blood sugar meters that are embedded under your skin without batteries. They rely on blood sugar and blood oxygen to react within a fuel-cell, generating the power to manage the meter!