I'm not sure if you're looking only for examples where *checking* properness/separatedness/etc. is easier using valuative criteria; this is an example where proving something is made easier. The example is liftability of arcs under proper birational morphisms.

A bit of setup: Say $X$ is a variety over a field $k$; an arc $\gamma$ on $X$ is a morphism $\gamma: \mathrm{Spec}\, k[[t]] \to X$. Write $X_\infty$ for the set of arcs on $X$ (in fact, $X_\infty$ carries a natural scheme structure, but we ignore this). The set of arcs on $X$ carries a great deal of information about the singularities of $X$, so it's of interest in birational geometry to understand how sets of arcs behave under proper birational maps; in particular, given $f:Y\to X$ a proper birational map, we want to know what the relation between $Y_\infty$ and $X_\infty$ is. Say that $f$ is an isomorphism over $X-Z$ for some closed subset $Z$ of $X$. "Most" arcs on $X$ don't have scheme-theoretic image contained in $Z_\infty$ (i.e., the morphism $\mathrm{Spec}\, k[[t]] \to X$ doesn't factor through $Z$). Write $Z_\infty$ for the set of arcs on $X$ that do factor through $Z$, and likewise $(f^{-1}(Z))_\infty$ for the arcs on $Y$ factoring through the (set-theoretic) preimage $f^{-1}(Z)$.

We claim that the proper birational morphism $f:Y\to X$ induces a bijection between $Y_\infty-(f^{-1}(Z))_\infty$ and $X_\infty-Z_\infty$, which should be thought of as a bijection away from a "measure zero" set. Using the valuative criterion for properness though, this is super easy to see: take $\gamma $ in $X_\infty-Z_\infty$. Since $\gamma$ is not in $Z_\infty$ the image of the generic point $ \mathrm{Spec}\, k((t)) \to \mathrm{Spec}\, k[[t]] \to X$ lies in the locus where $f$ is an isomorphism, so we can lift $\mathrm{Spec}\, k((t)) \to X$ to a map $ \mathrm{Spec}\, k((t)) \to Y$, giving a diagram
$\require{AMScd}$
\begin{CD}
\mathrm{Spec}\, k((t)) @>>> Y\\
@V V V @VV f V\\
\mathrm{Spec}\, k[[t]] @>>> X
\end{CD}
Now, the valuative criterion for properness of $f$ says exactly that there is a *unique* lift of $\gamma$ to an arc $\mathrm{Spec}\, k[[t]] \to Y$, giving the desired bijection.

Without using the valuative criterion, the other way I can think to prove this is to recognize $f:Y\to X$ as the blowup of $X$ at some ideal sheaf, and then work locally with the natural charts on the blowup; this isn't hard, but involves involves using the nontrivial fact that all proper birational maps are blowups of some ideal.

The irreducibility of the space of curves of given genus, Pub. Math. IHÉS 36 (1969), p. 75-109. $\endgroup$ – abx May 1 '19 at 4:14