The algebra $S$ is etale, i.e., smooth of relative dimension $0$. A proof, that $S$ is
flat over $R$ can be found in
see Proposition $1.3.5$ on page $17$. The proof uses induction, and the
following lemma: Let $S$ be an $R$-algebra. Suppose that $M$ is an $S$-module,
$f\in S$ an element such that multiplication by f is injective on $M \otimes k(m)$ for all
$m \in Max (R)$, and $M$ is flat over $R$. Then $M/fM$ is flat over $R$.
I hope this is more than just $Spec(S)$ is smooth over $Spec(R)$, hence flat.