From my reading of the cited paper, FLT is not really needed. The authors require a recursive sequence of curves $\langle C_i : i \in I \rangle$ for which the following holds:

(1) The function $i \mapsto |C_i({\mathbb Q})|$ is computable.

(2) There is a function (possibly required to be recursive) $B: \{ g \in \omega : g \geq 2 \} \to \omega$ so that for any $g \geq 2$ and any curve $X$ of genus $g$ over ${\mathbb Q}$ there are no nonconstant maps $C_i \to X$ or $X \to C_i$ for $i \geq B(g)$.

The Fermat curves conveniently satisfy these properties, but it is not so hard to build a sequence of curves over the rational numbers of growing genus for which local conditions preclude the existence of rational points and whose Jacobians are simple as abelian varieties (say, by taking the genus to be prime).

From my reading of the cited paper, FLT is not really needed. The authors require a recursive sequence of curves $\langle C_i : i \in I \rangle$ for which the following holds:

(1) The function $i \mapsto |C_i({\mathbb Q})|$ is computable.

(2) There is a function (possibly required to be recursive) $B: \{ g \in \omega : g \geq 2 \} \to \omega$ so that for any $g \geq 2$ and any curve $X$ of genus $g$ over ${\mathbb Q}$ there are no nonconstant maps $C_i \to X$ or $X \to C_i$ for $i \geq B(g)$.

The Fermat curves conveniently satisfy these properties, but it is not so hard to build a sequence of curves over the rational numbers of growing genus for which local conditions preclude the existence of rational points and whose Jacobians are simple as abelian varieties (Edit: comment about prime genus removed; instead, one needs to argue that the generic curve does have a simple Jacobian ).