I very much agree with your first point.
Many people don’t understand general relativity, and this is very prominent unfortunately in the astrophysics community, that mostly always sees things from a Newtonian perspective.
When people talk about GR working/not working, they mean solutions of the field equations that are then tested against observations. Einstein’s equations themselves simply relate curvature of spacetime to the energy-momentum tensor.
The current standard model of cosmology relies on spatial homogeneity and isotropy, which yields solutions of the EFE that have 6 Killing vectors/a six-dimensional isometry group. In addition, they also consider spatially flat models. To make “these” models fit with observations, they then introduce dark matter/energy.
I don’t believe a high degree of isotropy indicates isotropy. Something is either isotopic or it isn’t. Further, even from an astrophysical perspective, it is clear that the existence of black holes in the universe implies that the universe is inhomogeneous. There was a lot of work done on the Swiss Cheese models that were shown to challenge the dark energy assumptions. This also leads to the important matching / boundary conditions of how to join a S-metric universe with FLRW.