I know that GR must be diffeomorphism invariant, which (in my own words) means that GR, and by extension any observable, should not care about what coordinate system one chooses to use.
Suppose I have a spacetime metric that is fairly complicated when it is written in terms of coordinates ($t,r,\theta,\phi$). If this spacetime metric can be written in a simpler form under a change of coordinates (such as Minkowski, or FLRW), what is the significance this?
If the EFE are easier to solve and work with in these new coordinates that would be obviously useful, but I would still think that I need to convert back to the original coordinate system to describe what is going on in the universe described by the ($t,r,\theta,\phi$) directions.
I think my question is mainly (descriptively) how should I think about diffeomorphism invariance, when is it useful, and when is it not useful, and what does it tell us.
The second answer from this post helped, but I think that I do not really appreciating the utility of diffeomorphism invariance. The universe shouldn't care what coordinate system I choose to work with, even though I want to write predictions for observables in terms of coordinates ($t,r,\theta,\phi$).