> GR says that the Newtonian law of gravity is an approximation that makes reasonably accurate predictions when the spacetime curvature is small and all relative motions are slow compared to the speed of light.

These are merely the aforementioned caveats.

> It does not say that the Newtonian interpretation of that equation is correct.

If the interpretation were wrong, and that's not a force, then the amount of force in that equation would be 0.

With no force, Gm₁m₂/r² = 0.

However, that's not the modification that GR applies to this, though.