> The choice quote from part 1: "[our] final interpretation says that every spacetime is locally approximately flat in the sense that near any point of any spacetime (or near sufficiently small segments of a curve), there exists a flat metric that coincides with the spacetime metric to first order at that point (or on that curve) and approximates it arbitrarily well," [emphasis mine].

This statement is a mathematical conclusion from GR of a similar nature to noting that for any point on a sphere, there is a map projection onto the plane where distances on the sphere coincide to first order to distances on the plane.

This no more or less means that space-time is locally flat than it means that a sphere is locally flat. To a first order approximation, it is. But when we calculate the curvature tensor, we find that it isn't flat at all.