What are the generators of spherical symmetry?

The first thing it's important to understand is the notion of symmetry in general relativity. It is subtly different from the concept in Hamiltonian mechanics.

We say that a one parameter family of diffeomorphisms $\phi_t$ with velocity vector field $X$ preserves a tensor field $T$ iff

$$\mathcal{L}_X T = 0$$

If you aren't sure about the terminology above then read this handy intro. The vector field $X$ is often referred to as the generator of the transformations $\phi_t$.

We say that $X$ generates a symmetry of a metric spacetime if the associated $\phi_t$ preserve the metric $g$, or equivalently


This is exactly the condition that $X$ is a Killing vector field for the metric $g$. So the Killing vector fields are exactly those vector fields which generate spacetime symmetries.

Back to your example. In a spacetime with spherical symmetry you should be able to identify the Killing vectors above, using spherical polar coordinates.

Conversely, if (no subset of) the Killing vectors of your manifold obeys the Killing algebra of $S^2$ then you may conclude that your manifold doesn't have spherical symmetry. This is because the Killing vectors determine the Lie algebra of the maximal symmetry group preserving the metric, by definition.