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The 710 μF is relative to infinite distance. That's what you would see with a setup a million kilometers away and a cable going to Earth. If you have a setup in your lab then you never change the overall charge of Earth because that sum includes your lab. You can exchange charge with Earth with the only limit coming from your lab setup. It's a bit like to charging a capacitor in a circuit: You don't calculate the capacitance of the overall capacitor to infinity because all the charge re-distribution is internal to the capacitor.

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.

To add to this, the earth is also frequently not a large enough capacitor, especially for higher power applications. Large radio stations and radar stations often have a metal mesh placed on or under the ground nearby to make the “earth” a better capacitor. Lightning discharge spikes often extend deep into the foundations of buildings, or sometimes have a spike that extends below the foundations of the building, to get a better ground connection.

So the earth is often assumed to be infinite during the initial design stage, and if it turns out not to be “infinite” enough we can supplement the “earth” so that our design still works.

Just be careful with specific technical definitions of things.

A capacitance of 710 uF just means that if you charge up the earth's surface with 710 uC of charge then dragging an electron from infinity onto earth will impart 1 eV of energy to it. (One volt of potential from infinity)

That's not what you are doing when you are analyzing some circuit grounded to earth. If you throw a bucket of salt water onto some transformer or something and short some high voltage line to ground then you are actually just redistributing electrons and ions around within the earth's surface and in the atmosphere. You can do this "infinitely" because those electrons eventually will go back to / come back from wherever this high voltage is originating from.

However as a caveat there genuinely is a lot of voltage created from all these charge imbalances and therefore the capacitences of these systems is indeed quite small just like theoretical equations predict. Like the atmosphere has something like a few hundred volts per meter potential if you are going straight up and as I understand this huge potential is created by a relatively small numner of imbalanced ions in the atmosphere. Also if you actually fault a grounding connection in practice then you do get a large potentially dangerous amount of voltage on the soil.

Maybe this is wrong, but as I understand it the more correct model would be having a resistor connected to a small capacitor which is the earth, but this small capacitor has a discharge resistor that eventually drains all the voltage away as the charges return to the source.