Not really. For radial free fall geodesic, both t coordinate and r coordinate change as a function of an affine parameter. For a radial spatial geodesic, the t coordinate does’t change at all. Thus, they are completely curves in spacetime.
But they are two different curves. The latter is not even well defined as such - you have to specify that you mean radial path in an SC coordinate constant time slice. Then, the spatial the spatial path is a geodesic of both the slice and the spacetime. For other coordinates, a radial spatial path need not be a geodesic of the spacetime, though it would normally be a geodesic of the slice.- The world-line of a radial free fall is a geodesic in space-time
- The spatial-path of a radial free fall is a geodesic in space
The claim under discussion was a specific path being a geodesic of the spacetime and the submanifold. Your example is simply not relevant in that you are talking about two different paths.