Virtual critical regularity of mapping class group actions on the circle

We show that if $G_1$ and $G_2$ are non-solvable groups, then no $C^{1,tau}$ action of $(G_1times G_2)*mathbb{Z}$ on $S^1$ is faithful for $tau>0$. As a corollary, if $S$ is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group $mathrm{Mod}(S)$ with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.