How do fiber bundles help us to solve problems on a manifold? I'm trying to generate some intuition as to how the tangent bundle and cotangent bundle work in the context of Lagrangian and Hamiltonian dynamics.
Discussions on this topic that I've found so far in other fora (like physics.se and quora) seem to focus on "what it enables us to do" which I appreciate, but I'm looking for a more mathematical explanation about how they encode information about the manifold, i.e. "how it lets us do those things."
Here is what I think I know. Please assume the context of a finite dimensional smooth real manifold like you would use in mechanics.

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*A smooth curve in the manifold tells you the position an instantaneous velocity at each point, and seemingly tells you everything about the situation already. Lagrangian and Hamiltonian dynamics move the problem to a curve in the tangent/cotangent bundles where ostensibly it is easier to find the right curve. So the tangent/cotangent bundles "enrich" the manifold to aid solving the problem.

*At first it looked like the cotangent space was a refinement of an enrichment, but I've been getting a different impression now that they are simply different enrichments.  They have different qualities, and some of their functionals are paired naturally with the Legendre transformation.

Perhaps a categorical explanation would help me? To that end, I've looked at quite a few nLab articles, especially this one and this nLab article but that latter one is pretty advanced for me at the moment.
I didn't write out the details but it seems to me like there is a covariant functor from the category of differentiable manifolds into itself that takes a manifold to its tangent bundle and each morphism to its pushforward, and likewise a contravariant functor for cotangent bundles that uses the pullback, and I'm hoping the second link essentially confirms that.
So, given that background

How does passing to the tangent/cotangent bundles give us leverage as compared the original manifold?

 A: In the context of physics the point is that Newton's laws tell you that you need to know the initial position and momentum of a particle to predict its trajectory (let me stick to the cotangent bundle and Hamiltonian mechanics because I know how to say the thing I want to say for that case only). This is physical input; a priori physics could've turned out to work a different way, as far as I know.
So in other words, if you want to think of time evolution as a flow generated by a vector field on something, it's not a flow or a vector field on space $X$, because position is not enough information to determine time evolution. It is a flow / vector field on the cotangent bundle $T^{\ast}(X)$, because (position, momentum) is enough information to determine time evolution. And in fact time evolution is precisely the flow generated by the Hamiltonian vector field associated to the Hamiltonian.
Zooming out a little, Hamiltonian mechanics has to somehow take as input a smooth function $H$ (the Hamiltonian) and produce as output a vector field $X_H$ (time evolution); moreover $X_H$ must have the property that $L_{X_H} H = 0$ (conservation of energy). And if you sit down and write out what properties this implies about the pairing $\{ f, g \} = L_{X_f} g$ on smooth functions you'll write down the axioms of a Poisson bracket. Now $T^{\ast}(X)$ is canonically a symplectic manifold, so $C^{\infty}(T^{\ast}(X))$ canonically acquires a Poisson bracket defined by the symplectic form, and it turns out that Hamiltonian mechanics can be described entirely in terms of this bracket.
It is sometimes helpful to understand all of this as a classical limit of quantum mechanics. In quantum mechanics the Poisson bracket becomes a commutator bracket on a noncommutative algebra of observables, time evolution is given on observables by exponentiating a commutator which corresponds to an inner automorphism
$$A \mapsto e^{\frac{i H t}{\hbar}} A e^{- \frac{i H t}{\hbar}}$$
(this is the Heisenberg picture), and Noether's theorem reduces to the triviality that if $A$ and $H$ commute then $A$ is constant with respect to the above time evolution. From this point of view the significance of the cotangent bundle $T^{\ast}(X)$ is that it arises as the classical limit, suitably understood, of the noncommutative algebra of differential operators on $X$, and its Poisson bracket is a "commutative shadow" of this noncommutative origin. There are various versions of this story explaining how Poisson brackets show up as classical limits of commutator brackets, e.g. deformation quantization.
