@G.Bergeron Yes, which is nothing special to GR - most physical field theories are "diffeomorphism invariant". The special thing is that GR has the notion of a
covariant derivative and the gauge fields - Christoffels - that come with it. From a really "gaugey" viewpoint, someone took an abstract $\mathrm{GL}(n)$ gauge theory (not YM, obviously) and
soldered it