“Spacetime is curved” (and specifying how it is curved by writing the metric) is a much stronger statement than “there is spatial variation in the velocity of light” (with an appropriate quantification of that statement). The reason for this is that the statement about curved spacetime allows us to make predictions about how any physical process would proceed in this geometry by applying the equivalence principle, whereas the alternative statement only describes light propagation.
For example “curved spacetime” allows us to make a prediction that clocks would “slow down” near a massive body, regardless of the precise physical nature of time measurement performed by clock. We also can make prediction that neutrinos and other ultrarelativistic particles would move along nearly the same trajectories in this spacetime as light rays.
On the other hand, specifying “spatial variation in the velocity of light” would allow a physical realization of this law not only in the form of a curved spacetime but also via other analogue models of gravity (such as metamaterials). In a latter case, light propagation would formally be the same as in an equivalent gravitational field, but other physical processes would be different.
Another technical point to consider: providing the kinematics of light propagation would allow us to reconstruct only conformal structure of the spacetime and not its metric. So knowing coordinate velocities of light would not allow us to distinguish between the two metrics differing by a conformal factor.