I understand that writing the E-H action in terms of tetrads makes evident GR is a gauge theory. IOW general covariance/diffeomorphism invariance in GR is a form of gauge invariance. However unlike other gauge theories(for instance EM dependence on Minkowski spacetime), this gauge invariance in GR is accompanied by background independence(no prior global geometry/topology), wich makes us see GR's manifold abstractly as a simple differentiable manifold, that only has a casual geometric appearance when for practical requirements of the problem at hand like exploiting its symmetries we make a gauge choice and use some preferred coordinates(i.e. FRW coordinates in cosmology or Schwarzschild coordinates for isolated sources) but if GR is a gauge theory those gauge choices are not physical. Assuming this is not a wrong depiction of GR's general covariance, background independence and gauge invariance, a questions occurs to me: If GR is background independent, solutions of the EFE are valid locally like are all the observables derived from this local curvature geometry, but we shouldn't be able to infer any global geometry/topology from them, so I don't see any justification for "maximally extended solutions" like say, Kruskal-Szekeres solution. Being rigorous it seems there is no physical grounds to use it if we take seriously gauge invariance. How is this usually dealt with?