Neither you nor me are geometers so I would appreciate assistance on this point from an expert.No, we have an R2's worth of S2 surfaces, so to speak; that is, we have a 2-parameter family of S2 surfaces, each of which is an "intrinsic S2 manifold" in and of itself, and each one of which has the appropriate metric on it for spherical symmetry.
The way I understand the Schwarzschild spacetime topology S2XR2 is this, first of all one of the 4 dimensions is evidently time so we are left to study the spherical sy mmetry of the three dimensional hypersurface (the staticity of the manifold allows us to do this clea cut foliation). Agree so far?
We are left then with the direct product of the sphere with the real line(S2XR), unless you think it is R2XS1, but I don't think it is the case as it would lead to closed timelike curves (S1 time dimension).
Here is what I found online from a seminar of prof. Farkas on this:
"There are eight homogeneous simply connected geometries which give rise to compact three-manifolds. One of the simplest of the non-constant curvature ones is the space S2xR, which as its name suggests, is the direct product of the sphere with the real line. [....] in turn give rise to the 4 well-known compact manifolds admitting S2xR geometry. "
Now it is my understanding a constant curvature 3-manifold demands both homogeneity and isotropy, this space is homogeneous but doesn't have constant curvature.
I could argue about this but let's leave this aside for a moment as we would only really need to get clear on the above.The geodesic incompleteness doesn't have any effect on the spherical symmetry; it's a separate issue.