It seems to me that frequently when discussing stack conditions and descent, people consider only singleton covering families, i.e. there is some single covering map $U\to X$, for which one constructs a category of descent data out of $U$ and $U\times_X U$, and so on. But many sites have non-singleton covering families $(U_i \to X)_{i\in I}$; why are those ignored? Is there some deep reason why it suffices to consider singleton families?

My efforts to understand this so far have led me to think about "superextensive sites", which are sites whose covering families are generated by singleton covers together with inclusions into coproducts (the "extensive topology"). In particular, a fibered category is a stack on a superextensive site iff it is a stack for the singleton covers and also for the extensive topology. But while stack conditions for the extensive topology have an exceptionally simple form (the compatibility conditions being mostly vacuous), they are still not automatic. So why are they often not mentioned?