We first prove that we can find a $U$-admissible blowup such that the strict transform is flat. The question is \'etale local on the source and the target, see Lemma \ref{lemma-flatten-module-etale-localize} for a precise statement. In particular, we may assume that $S = \Spec(R)$ and $X = \Spec(A)$ are affine. For $s \in S$ write $\mathcal{F}_s = \mathcal{F}|_{X_s}$ (pullback of $\mathcal{F}$ to the fibre). As $X \to S$ is of finite type $d = \max_{s \in S} \dim(\text{Supp}(\mathcal{F}_s))$ is an integer. We will do induction on $d$. \medskip\noindent Let $x \in X$ be a point of $X$ lying over $s \in S$ with $\dim_x(\text{Supp}(\mathcal{F}_s)) = d$. Apply Lemma \ref{lemma-elementary-devissage} to get $g : X' \to X$, $e : S' \to S$, $i : Z' \to X'$, and $\pi : Z' \to Y'$. Observe that $Y' \to S'$ is a smooth morphism of affines with geometrically irreducible fibres of dimension $d$. Because the problem is \'etale local it suffices to prove the theorem for $g^*\mathcal{F}/X'/S'$. Because $i : Z' \to X'$ is a closed immersion of finite presentation (and since strict transform commutes with affine pushforward, see Divisors, Lemma \ref{divisors-lemma-strict-transform-affine}) it suffices to prove the flattening result for $\mathcal{G}$. Since $\pi$ is finite (hence also affine) it suffices to prove the flattening result for $\pi_*\mathcal{G}/Y'/S'$. Thus we may assume that $X \to S$ is a smooth morphism of affines with geometrically irreducible fibres of dimension $d$. \medskip\noindent Next, we apply a blowup as in Lemma \ref{lemma-flatten-module-pre}. Doing so we reach the situation where there exists an open $V \subset X$ surjecting onto $S$ such that $\mathcal{F}|_V$ is finite locally free. Let $\xi \in X$ be the generic point of $X_s$. Let $r = \dim_{\kappa(\xi)} \mathcal{F}_\xi \otimes \kappa(\xi)$. Choose a map $\alpha : \mathcal{O}_X^{\oplus r} \to \mathcal{F}$ which induces an isomorphism $\kappa(\xi)^{\oplus r} \to \mathcal{F}_\xi \otimes \kappa(\xi)$. Because $\mathcal{F}$ is locally free over $V$ we find an open neighbourhood $W$ of $\xi$ where $\alpha$ is an isomorphism. Shrink $S$ to an affine open neighbourhood of $s$ such that $W \to S$ is surjective. Say $\mathcal{F}$ is the quasi-coherent module associated to the $A$-module $N$. Since $\mathcal{F}$ is flat over $S$ at all generic points of fibres (in fact at all points of $W$), we see that $$ \alpha_\mathfrak p : A_\mathfrak p^{\oplus r} \to N_\mathfrak p $$ is universally injective for all primes $\mathfrak p$ of $R$, see Lemma \ref{lemma-induction-step}. Hence $\alpha$ is universally injective, see Algebra, Lemma \ref{algebra-lemma-universally-injective-check-stalks}. Set $\mathcal{H} = \Coker(\alpha)$. By Divisors, Lemma \ref{divisors-lemma-strict-transform-universally-injective} we see that, given a $U$-admissible blowup $S' \to S$ the strict transforms of $\mathcal{F}'$ and $\mathcal{H}'$ fit into an exact sequence $$ 0 \to \mathcal{O}_{X \times_S S'}^{\oplus r} \to \mathcal{F}' \to \mathcal{H}' \to 0 $$ Hence Lemma \ref{lemma-induction-step} also shows that $\mathcal{F}'$ is flat at a point $x'$ if and only if $\mathcal{H}'$ is flat at that point. In particular $\mathcal{H}_U$ is flat over $U$ and $\mathcal{H}_U$ is a module of finite presentation. We may apply the induction hypothesis to $\mathcal{H}$ to see that there exists a $U$-admissible blowup such that the strict transform $\mathcal{H}'$ is flat as desired. \medskip\noindent To finish the proof of the theorem we still have to show that $\mathcal{F}'$ is a module of finite presentation (after possibly another $U$-admissible blowup). This follows from Lemma \ref{lemma-flat-finite-type-finitely-presented-over-dense-open} as we can assume $U \subset S$ is scheme theoretically dense (see third paragraph of Remark \ref{remark-successive-blowups}). This finishes the proof of the theorem.