Let $k$ be an algebraically closed field. As the coarse moduli space of curves $M_g$ of genus $g$ over $k$ is not a fine moduli space, it does not have a universal family. But I am wondering if it has a family (proper and flat) such that the fiber over every point $[C]$ of $M_g$ is isomorphic to the curve $C$.

As a disclaimer: I am not that familiar with the language of stacks. As far as I understand the situation in this context, the stack $\mathcal{M}_g$ has an universal family $\mathcal{C}_g$. The corresponding coarse moduli space of $\mathcal{C}_g$ is $M_{g,1}$, so the coarse moduli space of curves with one marked point. The morphism $\pi \colon M_{g,1} \to M_g$ on the level of quasiprojective varieties is just forgetting about the marked point. This family has the property that the fiber over a point $[C]$ is isomorphic to $C$, at least if $C$ has no nontrivial automorphisms. In all other cases the fiber is isomorphic to $C/\operatorname{Aut}(C)$. Is it possible to get something better than that?