 Solution

The monodromy around both $0$ and $1$ must be a product of two disjoint transpositions, hence a $(2,2)$ cycle type in $S_4$. The product relation for loops around the three punctures says ${\sigma }_0{\sigma }_1{\sigma }_{\infty }=1$. By the stated fact, ${\sigma }_0{\sigma }_1$ is either the identity or another $(2,2)$ permutation. Consequently ${\sigma }_{\infty }=({\sigma }_0{\sigma }_1{)}^{-1}$ cannot contain a $3$-cycle. But ramification index $3$ over infinity would force exactly such a cycle, a contradiction.