Proper morphisms with geometrically reduced and connected fibers

Let $f: X to S$ be a proper morphism ($S$ locally noetherian), and $X to S’ to S$ its Stein factorisation. By Zariski’s Main Theorem the number of geometric connected components of the fibers of $f$ can be read from the cardinal of the fibers of the finite $S’ to S$. In particular if all fibers of $f$ are geometrically connected, then $S’ to S$ is radicial.

I expect that if furthermore the fibers of $f$ are geometrically reduced (and $f$ is surjective and $S$ reduced in order to remove trivial counterexamples), then $S’=S$ that is $f$ is an $mathcal{O}$-morphism (viz. $f_*mathcal{O}_X = mathcal{O}_S$). Strangely I only find this fact when $f$ is furthermore assumed to be flat, for instance: https://stacks.math.columbia.edu/tag/0E0L.

Here is an outline of a demonstration (suggested by a friend): we want to show that $S’ to S$ is an isomorphism. Since it is a surjection by assumption on $f$, it suffices to show that it is an immersion. By our assumptions on $f$, $S’ to S$ has geometrically connected and reduced fibers. We way assume that $S=textrm{Spec} A$ and $S’=textrm{Spec} B$, with $A to B$ finite. Let $C$ be the cokernel of $A to B$ (seen as a $A$-module). If $p$ is a prime ideal in $A$, $B otimes_A overline{k}(p) = overline{k}(p)$ (since it is connected and reduced over $overline{k}(p)$), so $C otimes_A overline{k}(p)=0$, so $C=0$.

Is the above proof indeed correct? Does the hypotheses already imply that $f$ is flat? Is there a reference to this result somewhere in the literature, presumably in EGA?