This has relatively little to do with the j-invariant itself. If you take any rational function $f(x)\in k(x)$ and let $G:=Gal(f(x) - t/k(t))$ (also referred to as the monodromy group of $f$), then by elementary Galois theory, $Gal(f(x)-f(y) / k(y))$ is a point stabilizer in $G$ (in the usual action on the roots). In your cases $G$ is $P\Gamma L_2(q)$ acting on $q+1$ points ($q=5,7,8,13$) which has solvable point stabilizer.
(Edit: Of course the reason why this observation "wasn't there" in the linked Jones/Roberts paper is that this is comparatively the "trivial case", whereas they are interested in (exceptional) cases where a group is a monodromy group of two essentially different rational functions in the same Galois closure.)