Define a topological category as a small category with topologies on the set of objects $C_0$ and on the set of arrows $C_1$, such that the domain map and the codomain map are continuous.
One can define a classifying topos $\mathcal BC$ of such a category, by taking topological sheaves on $C_0$ with a suitable action of $C_1$ (cfr. Moerdijk, Classifying spaces and classifying topoi).
What I would like to know is why topological categories are so important and what do they add to the context of Grothendieck topologies and Grothendieck sites.
In fact, I have asked elsewhere if, given a topological category $C$, one can build a site of definition for $\mathcal BC$ without already knowing that this latter is a topos. (Because in that case there is always the "Giraud" construction.) But I have obtained no answer.
So what I seek here is a historical contextualization and "qualitative" motivation. Why should one study topological categories in addition to Grothendieck sites?
Thank you in advance.