You write:
So, I'm wondering for some applications where I really need to use a bundle --- where some important fact is not true for general submersions (or, surjective submersions with connected fibers, say).
Actually, I am going to play devil's advocate here: sometimes it's better to have a submersion! This point comes up in a very relevant way in the classical smoothing theory of topological manifolds. Siebenmann (cf. Kirby and Siebenmann's book) defines a moduli space of smoothings of a topological manifold $M$ to be the space of $$(N,f)$$ such that $N$ is smooth and $f: N \to M$ is a homeomorphism.
Siebenmann chooses to topologize this in what seems a funny way: a $k$-simplex of such things is a pair $(N,f)$, where now $N \to \Delta^k$ is a smooth submersion (not necessarily proper if $M$ isn't compact!) and $f: N \to M \times \Delta^k$ is a homeomorphism. This gives a $\Delta$-space (a simplicial set w/o degeneracies). Call it's geometric realization $\text{Sm}(M)$.
Why doesn't he just topologize families as fiber bundles?
Here's why:
Let ${\cal O}_M$ be the poset of open subsets of $M$ which are abstractly homeomorphic to open balls. The fundamental theorem of smoothing theory asserts that the contravariant functor $\text{Sm} : {\cal O}_M \to \text{Top}$ given by $$ U \mapsto \text{Sm}(U) $$ is a "homotopy sheaf" if $\dim M \ge 5$, i.e., the (restriction) map $$ \text{Sm}(M) \to \underset{U \in {\cal O}_M} {\text{holim}}\quad \text{Sm}(U) $$ is a homotopy equivalence. This would not be the case if we had defined the families as bundles (rather than as submersions). Note: we cannot appeal to Ehresmann here as the submersions which are used in the define $k$-simplices in $\text{Sm}(U)$ are not assumed to be proper.