  We describe the $10$-dimensional space of $Sp(2)$-invariant $G_2$-structures
on the homogeneous $7$-sphere $S^7=Sp(2)/Sp(1)$ as $\mathbb{R}^+\times
Gl^+(3,\mathbb{R})$. In those terms, we formulate a general Ansatz for
$G_2$-structures, which realises representatives in each of the $7$ possible
isometric classes of homogeneous $G_2$-structures. Moreover, the well-known
nearly parallel round and squashed metrics occur naturally as opposite poles in
an $S^3$-family, the equator of which is a new $S^2$-family of coclosed
$G_2$-structures satisfying the harmonicity condition $div T=0$. We show
general existence of harmonic representatives of $G_2$-structures in each
isometric class through explicit solutions of the associated flow and describe
the qualitative behaviour of the flow. We study the stability of the Dirichlet
gradient flow near these critical points, showing explicit examples of
degenerate and nondegenerate local maxima and minima, at various regimes of the
general Ansatz. Finally, for metrics outside of the Ansatz, we identify
families of harmonic $G_2$-structures, prove long-time existence of the flow
and study the stability properties of some well-chosen examples.
