> it's not every day that we extend models into a domain (near the singularity) where we know our model must eventually break down somehowYes, but that doesn't change what the model says. It just affects how likely we think it is that the model is actually realized in our universe. I agree that it's quite likely that the standard black hole model I described isn't realized in our actual universe. But we can still use it if we don't have any better model to replace it with, since even if it breaks down near the singularity, that still leaves the whole rest of the model with plenty of usefulness.
What's of great interest about alternate models for collapsed objects that have dark energy inside, like the Bardeen "black hole", is that they do hold out the promise of being a better model to replace the standard black hole model, that doesn't have a singularity anywhere and so would not be expected to break down the way we think the standard black hole model breaks down near the singularity.
> Is the claim that the matter simply disappears from the spacetime, with all its conserved quantum numbers and other conserved quantities?
The matter disappears, but conserved quantities do not. They remain embedded in the spacetime geometry that is left behind.
> The Singularity Theorems assume the existence of a trapped surface.
Yes, but the alternate "black hole" models with dark energy inside, such as the Bardeen "black hole", also have trapped surfaces, so this doesn't help to distinguish the models.
The singularity theorems also assume energy conditions. Those are the conditions that the models with dark energy inside violate, and which allow those models to not have singularities even though they do have trapped surfaces.
> Is there a clear proof that in realistic models of black hole formation we must have a trapped surface eventually?
I don't know about "proof", but there are plenty of numerical simulations of realistic collapses of massive objects like stars that show trapped surfaces forming. So I would say it's a robust expectation of any such collapse process, even if we don't have an ironclad proof that it must occur in every single case.
> the Nobel committee dubiously claimed that Penrose's Singularity Theorems prove that black hole formation and singularities "are a robust prediction of the general theory of relativity".
That statement was justified. But saying that it's a robust prediction if particular conditions are satisfied is not the same as saying that all of those conditions must be satisfied in our actual universe.