This is meant to be a modest complementary of naf and Evgeny Shinder's answer to this question, showing that 
the Main Proposition (over $\mathbf{C}$) 
also follows from Kodaira's work of elliptic surfaces.

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.