[First proof] This proof is longer but does not use the existence of a complete d\'evissage. The problem is local around $x$ and $s$, hence we may assume that $X$ and $S$ are affine. During the proof we will finitely many times replace $S$ by an elementary \'etale neighbourhood of $(S, s)$. The goal is then to find (after such a replacement) an open $V \subset X \times_S \Spec(\mathcal{O}_{S, s})$ containing $x$ such that $\mathcal{F}|_V$ is flat over $S$ and finitely presented. Of course we may also replace $S$ by $\Spec(\mathcal{O}_{S, s})$ at any point of the proof, i.e., we may assume $S$ is a local scheme. We will prove the proposition by induction on the integer $n = \dim_x(\text{Supp}(\mathcal{F}_s))$. \medskip\noindent We can choose \begin{enumerate} \item elementary \'etale neighbourhoods $g : (X', x') \to (X, x)$, $e : (S', s') \to (S, s)$, \item a commutative diagram $$ \xymatrix{ X \ar[dd]_f & X' \ar[dd] \ar[l]^g & Z' \ar[l]^i \ar[d]^\pi \\ & & Y' \ar[d]^h \\ S & S' \ar[l]_e & S' \ar@{=}[l] } $$ \item a point $z' \in Z'$ with $i(z') = x'$, $y' = \pi(z')$, $h(y') = s'$, \item a finite type quasi-coherent $\mathcal{O}_{Z'}$-module $\mathcal{G}$, \end{enumerate} as in Lemma \ref{lemma-elementary-devissage}. We are going to replace $S$ by $\Spec(\mathcal{O}_{S', s'})$, see remarks in first paragraph of the proof. Consider the diagram $$ \xymatrix{ X_{\mathcal{O}_{S', s'}} \ar[ddr]_f & X'_{\mathcal{O}_{S', s'}} \ar[dd] \ar[l]^g & Z'_{\mathcal{O}_{S', s'}} \ar[l]^i \ar[d]^\pi \\ & & Y'_{\mathcal{O}_{S', s'}} \ar[dl]^h \\ & \Spec(\mathcal{O}_{S', s'}) } $$ Here we have base changed the schemes $X', Z', Y'$ over $S'$ via $\Spec(\mathcal{O}_{S', s'}) \to S'$ and the scheme $X$ over $S$ via $\Spec(\mathcal{O}_{S', s'}) \to S$. It is still the case that $g$ is \'etale, see Lemma \ref{lemma-etale-at-point}. After replacing $X$ by $X_{\mathcal{O}_{S', s'}}$, $X'$ by $X'_{\mathcal{O}_{S', s'}}$, $Z'$ by $Z'_{\mathcal{O}_{S', s'}}$, and $Y'$ by $Y'_{\mathcal{O}_{S', s'}}$ we may assume we have a diagram as Lemma \ref{lemma-elementary-devissage} where in addition $S = S'$ is a local scheme with closed point $s$. By Lemmas \ref{lemma-devissage-finite-presentation} and \ref{lemma-devissage-flat} the result for $Y' \to S$, the sheaf $\pi_*\mathcal{G}$, and the point $y'$ implies the result for $X \to S$, $\mathcal{F}$ and $x$. Hence we may assume that $S$ is local and $X \to S$ is a smooth morphism of affines with geometrically irreducible fibres of dimension $n$. \medskip\noindent The base case of the induction: $n = 0$. As $X \to S$ is smooth with geometrically irreducible fibres of dimension $0$ we see that $X \to S$ is an open immersion, see Descent, Lemma \ref{descent-lemma-universally-injective-etale-open-immersion}. As $S$ is local and the closed point is in the image of $X \to S$ we conclude that $X = S$. Thus we see that $\mathcal{F}$ corresponds to a finite flat $\mathcal{O}_{S, s}$ module. In this case the result follows from Algebra, Lemma \ref{algebra-lemma-finite-flat-local} which tells us that $\mathcal{F}$ is in fact finite free. \medskip\noindent The induction step. Assume the result holds whenever the dimension of the support in the closed fibre is $< n$. Write $S = \Spec(A)$, $X = \Spec(B)$ and $\mathcal{F} = \widetilde{N}$ for some $B$-module $N$. Note that $A$ is a local ring; denote its maximal ideal $\mathfrak m$. Then $\mathfrak p = \mathfrak mB$ is the unique minimal prime lying over $\mathfrak m$ as $X \to S$ has geometrically irreducible fibres. Finally, let $\mathfrak q \subset B$ be the prime corresponding to $x$. By Lemma \ref{lemma-induction-step} we can choose a map $$ \alpha : B^{\oplus r} \to N $$ such that $\kappa(\mathfrak p)^{\oplus r} \to N \otimes_B \kappa(\mathfrak p)$ is an isomorphism. Moreover, as $N_{\mathfrak q}$ is $A$-flat the lemma also shows that $\alpha$ is injective and that $\Coker(\alpha)_{\mathfrak q}$ is $A$-flat. Set $Q = \Coker(\alpha)$. Note that the support of $Q/\mathfrak mQ$ does not contain $\mathfrak p$. Hence it is certainly the case that $\dim_{\mathfrak q}(\text{Supp}(Q/\mathfrak mQ)) < n$. Combining everything we know about $Q$ we see that the induction hypothesis applies to $Q$. It follows that there exists an elementary \'etale morphism $(S', s) \to (S, s)$ such that the conclusion holds for $Q \otimes_A A'$ over $B \otimes_A A'$ where $A' = \mathcal{O}_{S', s'}$. After replacing $A$ by $A'$ we have an exact sequence $$ 0 \to B^{\oplus r} \to N \to Q \to 0 $$ (here we use that $\alpha$ is injective as mentioned above) of finite $B$-modules and we also get an element $g \in B$, $g \not \in \mathfrak q$ such that $Q_g$ is finitely presented over $B_g$ and flat over $A$. Since localization is exact we see that $$ 0 \to B_g^{\oplus r} \to N_g \to Q_g \to 0 $$ is still exact. As $B_g$ and $Q_g$ are flat over $A$ we conclude that $N_g$ is flat over $A$, see Algebra, Lemma \ref{algebra-lemma-flat-ses}, and as $B_g$ and $Q_g$ are finitely presented over $B_g$ the same holds for $N_g$, see Algebra, Lemma \ref{algebra-lemma-extension}.