To give an example of a peculiar feature of simplicial sets that I cannot remember encountering anywhere in the context of homotopy theory: every simplicial set $X$ possesses partial map classifier $X\rightarrowtail\widetilde X$: the $n$-simplices of $\widetilde X$ are partial simplices of $X$, i. e. maps from all kinds of simplicial subsets of the standard $n$-simplex $\Delta[n]$ to $X$. This $\widetilde X$ is a contractible Kan complex, in strongest possible way: it is an injective object, i. e. any $Y\leftarrowtail Y'\to\widetilde X$ extends to $Y$. This construction is functorial, in fact, part of a monad structure (sometimes called "lift monad" or "maybe monad"). Is not existence of such a thing useful for homotopy-theoretic purposes?

Note that this is just one example, there surely are many other features that can be extracted from the topos structure. I found a related question Internal logic of the topos of simplicial sets but it is rather about peculiarities of simplicial sets as a particular topos than peculiarities of this topos as a particular homotopy-theoretic universe.