Section 4.1.1 of Vistoli's Notes on Grothendieck topologies, fibered categories and descent theory motivates the formalism of stacks via topological spaces.
Proposition 4.1 says that given an open cover $(U_i\to U)$, every family of bundles $(X_i\to U_i)$ equipped with transition isomorphisms satisfying the cocycle relations gives rise to a bundle $X\to U$ which pulls back to $X_i\to U_i$ along $U_i\to U$.
The remark following proposition 4.1 says that the fact we may glue bundles as well as continuous maps into a fixed codomain means the codomain fibration of topological spaces is a stack.
I think the words "open cover" are not relevant to proposition 4.1: a family of bundles over the $U_i$ with transition isomorphisms satisfying the cocycle condition always yields a bundle over $U$ for any family $U_i\to U$, by the same proof.
I don't see how proposition 4.1 implies the codomain fibration of spaces is a stack. I think the key result is that singleton covers associated to open covers are effective descent morphisms, i.e that the category of bundles over $U$ is equivalent to the category of descent data. The fact bundles merely glue together does not mean all bundles over $U$ arise this way.
Am I misunderstanding something?
Indeed Proposition 4.1 holds for an arbitrary family of bundles. The proof is the same.
Indeed Proposition 4.1 does not imply the codomain fibration of topological spaces is a stack. The stack property depends on properties of open covers. Without them, the induced functor to the category of descent data will not be an equivalence.
As suggested in the comments, the example of covering a space $U$ by its family of points is instructive in demonstrating how the canonical functor may be faithful and even essentially surjective without being full (without the possibility of gluing bundle arrows).