We generally don't use capacitors to power devices because it discharges most of it's energy at the quickly unlike a battery which as a slower mostly constant output. It's useful if you want to dump a lot of energy at once like a camera flash or defibrillator, not so much when you want to power a motor.

The 710uF is the capacitance with a conductor at infinite distance, for any circuit you build on Earth the actual capacitance will be much much larger.

Much of electrical engineering is about simplifying circuit calculations. Earth ground could be any source/sink of charge sufficiently large as to be effectively infinite in the context of your circuit. And Earth is effectively infinitely large compared to most electronics at human scales.

Imagine every circuit connected to earth ground as one terminal of a capacitor and the Earth as the other terminal. Each circuit could have a different amount of current flowing to earth ground with a non zero net current. But because the capacitance is so big, V=Q/C, any amount of human scale current we provide will have negligible effects on the voltage of earth ground.

So you are right, Earth is not a infinite source/sink but we treat it as one because it is large enough to not impact our calculations most of the times.