The Gravity Assist allows for a spacecraft to gain more speed and energy ("stealing" it from the planet).
Can this approach be used for creating power plants that generate "cheap" energy for use on Earth?
Yes, you can.
It is entirely possible to create tidal power plants, stealing energy from the Earth-Moon system.
Eventually this source runs out of energy, but it would be a much longer duration than the duration until we run out of fossil fuels (or out of room to dump the carbon dioxide somewhere). So it could be considered renewable, like solar power is considered renewable (which it isn't since the sun runs out of fuel someday).
The spacecraft gain energy mainly because it do not "return" back in the place where the gravitational effect is applied, otherwise the "slingshot effect" would not be effective. So, in the earth, you can use "potential energy" when you have a differential height, but if you want to return to the initial position you must spend the same amount of energy to close the loop. Tidal energy is an exception, because the system sun-moon-earth is creating and subtracting energy to the sea in form of "waves", and you can steal energy on each front (rising or falling) of the waves, giving the sun-moon-earth is not returning on the same exact position as before after each wave cycle.