Most textbooks define a vector field on a smooth manifold $M$ as a section of the tangent bundle of $M$. My question is: why is it even necessary to talk about bundles when defining vector fields on $M$?
To be clear, when I refer to the tangent bundle, I refer to the triple $(M, TM, \pi)$, where $\pi: TM \to M$ is a smooth surjection. The problem is, vector fields can be defined without ever referring to $\pi$ at all. We can just define a smooth vector field as a smooth map $f: M \to TM$ satisfying $f (p) \in T_pM$ for all points $p \in M$ (that is, once we've given $TM$ a topology and a smooth atlas).
I'm likely missing something about the importance of bundles. Any comments would be most welcome. Thanks!