Obstructions to applying the Baker--Bilu method for determining integral points on curves

We prove that for every smooth projective integral curve $X$ of genus at least $2$ over $\mathbb C$, there exists $x \in X(\mathbb C)$ such that no connected finite \'etale cover of $X-\{x\}$ admits a nonconstant morphism to $\mathbb G_m$. This has implications for the applicability of Baker's method to determining integral points on curves.