Representation stability in the cohomology of configuration spaces

One of the achievements of the of the theory of representation stability was to exhibit this phenomenon in the cohomology of configuration spaces of connected manifolds of dimension at least 2 by Church, further clarified and improved by the framework of FI-modules by Church, Ellenberg, Farb, and Nagpal. There have been generalizations of this result which applies to more singular spaces (rather than only manifolds) by Tosteson and by Petersen.

I will outline yet another proof I have found of the stability statement for manifolds, open to further generalizations. The nontrivial ingredients will be
(1) Cohen's computation of the cohomology of configuration spaces of R^n, n > 1
(2) the existence of a certain spectral sequence about open coverings.

I will briefly talk about (1), and (2) can be treated as a black box. Modulo (1) and (2), the topological part of the argument uses only a little more than what is already in the definition of a manifold: locally Euclidean and Hausdorff!