For convenience let's focus on an isotropic ferromagnet, $H = -\sum \boldsymbol{S_i \cdot S_j}$.
On a classical level we are often given the picture of spin waves as slowly rotating spins, each sufficiently close to perfect alignment as to keep the energy cost really small. This is also the intuition given for Goldstone modes: one can use the continuous symmetry to start with a given spin direction and then apply bigger and bigger rotations as one goes further along.
However the quantum-mechanical picture seems completely different to me. There a spin wave with momentum $\boldsymbol k$ is created by applying the operator $\sum e^{-i \boldsymbol{k \cdot r}} \; S^+_\boldsymbol{r}$ on a ground state of all spins down. In other words it is a massive superposition where each state only has a single perturbed spin.
Self-contained, I understand the logic of each picture separately (the second having to do with thinking of how the Heisenberg Hamiltonian induces `spin hopping' etc) but I really can't see how the first picture arises from the latter, even in a large-$S$ limit. Can anyone clarify this for me?