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The action is not transitive on simple closed curves. For instance, some of the simple closed curves lift to the following unbranched, degree $2$ cover, yet others do not: $\text{Zero}(y^2-x(x-1)(x-a)(x-b)(x-c), x-z^2) \subset \mathbb{C}^3$. 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 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.