Here I essentially am just repeating what Siebenmann wrote in Essay V of the Kirby-Siebenmann volume while fetching some definitions from early sections:

Let $n \geq 5$. By work of Kirby and Siebenmann on the connectivity of $Top(n)/O(n) \rightarrow Top/O$, $\pi_n(Top/O)=\pi_n(Top(n)/O(n))$. The latter is what we will choose to work with. In section 5.3 the space $\Omega^n Top(n)/O(n)$ was identified with the space of smoothings of $S^n$ relative to a disk with standard smooth structure. Hence, $\pi_n(Top(n)/O(n))$ is the concordance classes of such smooth structures.

We can consider the inclusion of such smoothings into the homotopy smoothings of $S^n$ relative to a disk (perhaps up to h-cobordism?). Surjectivity follows from the Poincare conjecture and injectivity from what Siebenmann calls "Alexander isotopy", I am not sure what this is.

Then this set of homotopy smoothings of $S^n$ relative to a disk has a map to the group of oriented homotopy spheres given by using the homotopy equivalence to transport the orientation of $S^n$ to the homotopy sphere and then forgetting the homotopy equivalence. This map is surjective since any homotopy sphere contains a standard disk in it. The main thing to showing injectivity is that up to orientation there is a single isotopy class of embedded disks in our homotopy sphere.

This chain of bijections shows that $\pi_n(Top/O)$ for $n\geq 5$ is $\Theta_n$, the group of oriented homotopy spheres. I think any subtlety in this proof would arise from showing that including smoothings into homotopy smoothings is injective, so I will look further into "Alexander isotopy".