The action is not transitive on simple closed curves.  For instance, some of the simple closed curves lift to the following unbranched, $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ cover, yet others do not: the projective smooth model of the affine curve $\text{Zero}(y^2-x(x-1)(x-a)(x-b)(x-c), 4xz^2-(z^2+1)^2) \subset \mathbb{C}^3$.  Since your parameterization of the curves "fixes" the branch points over $x=0$, $x=1$, and $x=\infty$, you are also "fixing" the quotient fundamental group of the orbifold with underlying manifold $\mathbb{CP}^1$ (the "$x$-line") and $\mathbb{Z}/2\mathbb{Z}$-orbifold points at $x=0$, $x=1$, and $x=\infty$.  This orbifold fundamental group is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.  So the image of a free homotopy class in this quotient group is fixed by the monodromy action.