Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance? 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 $Y$. It is thus some sort of a fibrant cone for $X$. 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.
 A: To answer the title question: Yes, the fact that the 1-category $sSet$ is a 1-topos has homotopy-theoretic significance. It is closely related to the fact that the $\infty$-category $Spaces$ is an $\infty$-topos! To a large degree, constructions using the topos structure on $sSet$ can be recast model-independently as constructions using the $\infty$-topos structure on $Spaces$. If you like, I suppose you can see the fact that $Spaces$ is an $\infty$-topos as a consequence of the fact that $sSet$ is a 1-topos using Rezk's model topos machinery.
On the other hand, I'm not so sure about the significance of the partial map classifier in particular. The "$\infty$-partial map classifier" exists in $Spaces$, but there are not so many partial maps in the $\infty$-topos $Spaces$ as in the 1-topos $sSet$ (since a monomorphism is just a coproduct inclusion in $Spaces$). Probably this construction has some homotopy-theoretic significance in some model-dependent context, but it will depend on what the other model-dependent particulars of the situation are, I suspect.
On the gripping hand, an $\infty$-topos like $Spaces$ actually has object classifiers which are arguably even "better" than subobject classifiers.
