Different perspectives on smoothing theory

Question

To my understanding there are 3 common ways to phrase the fundamental result of smoothing theory (all dimensions will be $>4$):

1. Let $X$ be an $n$-dimensional manifold, then concordance classes of smooth structures on $X$ are in bijection with sections of the bundle $\Xi \times_{Top(n)} Top(n)/O(n)$ up to vertical homotopy, where $\Xi$ is the microbundle of $X$.

2. Let $X$ be an $n$-dimensional manifold, then picking a Serre fibration to represent $\rho: BO \rightarrow BTop$ (being careful to note these are stable versions), smooth structures on the underlying manifold $X$ up to concordance correspond to lifts of the stable microbundle of $X$ to $BO$.

3. Let $M$ be an $n$-dimensional manifold that admits a smoothing. Smooth structures on $M$ up to concordance are in correspondence with $[M,Top/O]$.

At first glance these all seem reasonable, at least when one knows that $Top(n)/O(n)$ is the space of smooth structures of $\mathbb{R}^n$. However, it gets weird (at least to me) when one looks closer and tries to relate these.

Number 1 is fine, it is formalizing the notion that smooth structures are patched together from smooth structures on the local neighborhoods of a manifold.

Number 2 is where it gets weird. The first thing to note is that we have actually stabilized the classifying spaces. Of course this is arising somehow from the product structure theorem which asserts and equivalence between smooth structures on $X$ and $X \times I$. So it appears that we are doing somethings like picking a vector space structure on our microbundle, and then using the fact that we can move up in dimension to argue this actually corresponds to a smooth structure, and then moving back down by the product structure theorem

Number 3 seems to be of importance because it is homotopically a simple way to phrase it. $Top/O$ is the homotopy fiber of $BO \rightarrow BTop$ and so if I pick a smooth structure on $M$, I can take its tangent bundle $\phi$ and consider any of the lifts from $2$ minus $\phi$ which should have a canonical nullhomotopy after being pushed down to $BTop$. This what the homotopy fiber classifies, so we get a map.

So here are some concrete questions:

1. Is it correct that the first and second correspondence is canonical while the third depends on a choice of smoothing?

2. Could someone provide a sketch of the second correspondence that is more detailed than mine? I have difficulty finding a reference that actually explains it.

3. Is there an easy way to relate the first and second correspondence?