This is what every flywheel already does (unless it is balanced by a second flywheel going the other way). Conservation of angular momentum: as a flywheel gains angular momentum, it must be compensated by the Earth gaining the same amount of angular momentum in the opposite direction.

Usually this isn't referred to as "spinning up the Earth" though :-)

If that was easy and cheap to do, our problem wouldn't be climate change due to greenhouse gases - it would be climate change due to the earth rotating slower since we'd been using it as an energy source for 200 years!

You'd speed it back up in times of energy surplus. It's a giant flywheel.

Rotational energy is already harvested using tidal power and wind power.

There are band type "windmills" it's a taught band with magnets on each end when it flaps it moves the magnets to generate power.