To me, Hausdorff metric is an unaccustomed way of making such a space of spaces. I think I don't trust it because fixing a homotopy type gives you a set that is neither closed nor open in general.
But yes I believe the picture is that some kind of "space of spaces of homotopy type " is closely related to .
Let's start with smooth manifolds, but of codimension zero. For a fixed and a finite complex , let be the space of smooth compact -manifolds containing in the interior as a deformation retract. (Let's say, the simplicial set where a -simplex is a suitable thing in such that the projection to is a smooth fiber bundle.) You can map by crossing with (and doing something about corners), and you can consider the (homotopy) colimit over . Using the classification of -cobordisms you can work out that the set of components is the Whitehead group of . The loopspace of one component is the smooth stable pseudoisotopy space of . To get the idea, think of the case when is a point: the space is then, after you discard extraneous components corresponding to cases where the boundary is not simply connected -- which were going to go away anyway upon stabilizing over -- your quotient of {embeddings }~ by {diffeomorphisms }. It's also a kind of "space of all -cobordisms on , and thus a delooping of the (unstable) pseudoisotopy space of .
When is more complicated than a point, it's important to distinguish between the space of all blah blah blah containing as a deformation retract and the space of all blah blah homotopy equivalent to ; they differ by the space of homotopy equivalences .
There is a similar story for the piecewise linear or topological case.
The piecewise linear manifold version of this construction can, I believe, be shown to be equivalent to a non-manifold construction more like what you asked about: some kind of "space of compact PL spaces in containing a fixed as deformation retract".
Waldhausen tells us that the stable smooth construction above and the stable PL construction above are respectively (the underlying spaces of spectra which are) the fiber of a map from the suspension spectrum of to , and the fiber of a map from to .
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