The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with duals upon a single object.
The proof sketch by Lurie uses Morse theory extensively, and has since been formalised and extended by several authors. Morse theory is a technique from real differential topology, and it is related to handle decompositions.
Piecewise linear (PL) manifolds are not equivalent to smooth manifolds from dimension 5 upwards. Topological manifolds famously already depart from PL and smooth manifolds in dimension 4. I don't know whether PL manifolds or topological have a well-developed analogue of Morse theory, but they have handle decompositions (except for 4d topological manifolds), so I would expect a lot of the technical procedures from the bordism hypothesis proof to work. I do not understand whether it is possible to define a $(\infty,n)$-category of bordisms in these cases, as opposed to a mere 1-category.
Then the awkward question is: How do the $(\infty,n)$-categories of PL and topological bordisms look like? They cannot possibly be equivalent to the smooth one, or otherwise we would have trivially proven that extended TQFTs do not detect PL or smooth structures? But then, which part of the construction goes awry? What's so special about the smooth categories that the bordism hypothesis works there?