I think I can now reasonably point to the book
Bourbaki, Topologie Algébrique: Chapitres 1 à 4 (Elements de Mathématique) 7 Apr 2016
which does use ideas of descent and the fundamental groupoid both for the van Kampen theorem and for discrete groups acting properly on spaces. They call the fundamental groupoid the "Poincaré groupoid", though I think its topological use goes back only to Reidemeister's 1932 book.
Their Theorem 3 on p. 419 seems to overlap with the work on orbit groupoids of Chapter 11 of Topology and Groupoids, which gives the application to the symmetric square of a space.
Comments welcome!