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I'm too lazy to do the math, but what's the point-to-point distance (through the planet) of points that are 5000km along the surface?

Assuming a spherical earth, it's 4873 km.

Assuming a spherical earth in a vacuum... One of the few times that simplifying assumption matches reality.

Not quite; the Earth is an oblate spheroid. Assuming it's a sphere is certainly a close enough approximation for this exercise, though.

How far is it if you assume a flat earth?

5000km