Edit notice: As Evgeny Shinder pointed out in the comment, under the assumption of the main proposition we don't have $S = J^{-1}(\{0,1,\infty\})$ but only $S = J^{-1}(\infty)$ where $J : C \to \mathbf{P}^1$ is the $j$-invariant map. The conclusion is now weaker than the original answer.
This is meant to be a modest complementary of naf and Evgeny Shinder's answer to this question, showing that a weaker version of the main proposition (over $\mathbf{C}$) also follows from Kodaira's work of elliptic surfaces. We will prove it under the stronger assumption that $S$ parameterizes exactly fibers of periods $0$ or $1$ and singular fibers.
We use the notations set up by Evgeny Shinder. As a consequence of the classification result summarized in BHPV's compact complex surface (Theorem V.11.1), the $j$-invariant map $J : C \to \mathbf{P}^1$ determines the elliptic surface $f: X \to C$ with a section up to finite ambiguity. It remains to show that there exist at most finitely many surjective maps $J : C \to \mathbf{P}^1$ such that $S = J^{-1}(S')$ where $S' := \{ 0,1,\infty \}$.
We first show that the degree $d$ of such a map $J : C \to \mathbf{P}^1$ is bounded. Indeed, if $R$ denotes the ramification degree of $J$, then since $S = J^{-1}(S')$, we have $$3d- |S| \le R$$ and Riemann-Hurwitz shows that $$2g(C) - 2 = -2d + R \ge d - |S|.$$ Hence $d \le 2g(C) - 2+ |S|$.
If we fix $g : S \to S'$, then the moduli space $\mathrm{Hom}(C,\mathbf{P}^1;g)$ of such maps $J : C \to \mathbf{P}^1$ satisfying the additional condition that $J_{|S} = g$ has tangent space at $J$ isomorphic to $H^0(C,J^*T_{\mathbf{P}^1} \otimes I_{S})$. As $S = J^{-1}(S')$ and $J$ is surjective, we have $\deg (J^*T_{\mathbf{P}^1} \otimes I_{S}) < 0$, and it follows from the boundedness of the degree that $\mathrm{Hom}(C,\mathbf{P}^1;g)$ is finite. Since there are only finitely many choices of $g$, the finiteness of the elliptic surfaces in question follows.