I feel like there should be better examples than the following (e.g., using matrix multiplication, which I tried in many variations).  However, this is the best I can see.  

Let $k$ be a field. Let $S$ be a $3$-dimensional vector space, $V \cong \text{Spec}k[x,y,z]$.  Thus $\mathbb{P}(V)$ is isomorphic to $\mathbb{P}^2$.  Let $\overline{X}$ be the closed subscheme of $V\times \mathbb{P}(V)$ parameterizing pairs $(v,[w])$ such that $v$ is a multiple over $w$, possibly zero.  In coordinates, $((x,y,z), [u,v,w])$, this is the zero scheme of $xv-yu, xw-zu, yw-zv$.  Denote by $\overline{\pi}:\overline{X} \to S$ the natural projection.  

Fix a line $L\subset \mathbb{P}(V)$, e.g., the zero scheme of $u$, and let $X$ be the open complement in $\overline{X}$ of $\{0\}\times L$.  Let $\pi:X\to S$ be the restriction of $\overline{\pi}$ to $X$.  Let $M$ be the structure sheaf of $X$.  This is not $S$-flat.  The universal flatification of $\overline{\pi}$ is $\overline{\pi}$, and the inverse image of the origin is $E\cong \mathbb{P}(V)$.  Certainly the strict transform of $\pi$ with respect to $\overline{\pi}$ is flat.  

However, I can take any finite collection of points inside $L$ inside $E$, I can "pinch" $\overline{X}$ along those points. That scheme maps to $X$, and that will also be a flatification.  What I would "like" to do in order to make a universal (proper, birational) flatification is contact /  $L$ inside $E$ inside $\overline{X}$.  But that is not possible: we cannot contract a line $L$ in a $2$-plane $E$ (not as a scheme, not even as an algebraic space).  Thus, there is no universal (proper, birational) flatification.