Ibix said:
I find it interesting that the collapsing matter doesn't seem to explain the singularity.
I'm not sure what you mean. The collapsing matter satisfies the premises of the classic singularity theorems, just as any closed matter-dominated FRW spacetime does.
Ibix said:
The singularity and the event horizon remain properties primarily of the Schwarzschild vacuum spacetime.
In the sense that "most of" the singularity lies in the vacuum region and not the collapsing matter, yes, this is true. But that doesn't change the fact that both regions are valid solutions of the EFE and both end in a singularity. The two regions, including the way they end in singularities, still have to match at the boundary between them.
Ibix said:
And it would still be there (I think) if we put in by hand some mechanism that stopped the matter collapsing at some ##r<R_S##.
There is no such mechanism possible. That's one of the implications of the singularity theorems: any surface with ##r < R_S## is a trapped surface, and the presence of a trapped surface, given that the other requirements for the singularity theorems are satisfied, is a sufficient condition for the collapsing matter to end in a singularity. The singularity theorems make no assumptions whatever about the details of the collapse; you can add whatever pressure, shock waves, etc. you want, and that won't change the fact that, once you have a trapped surface, a singularity is inevitable.
What
would change that outcome is for the equation of state of the "matter" to change in a way that violated the energy conditions. There are models in the literature that do this, for example the models that were discussed in a recent thread on the Bardeen black hole and models derived from it. In those models, the deep interior has a de Sitter-like geometry, which is one possibility for what quantum fields might do under conditions of strong enough spacetime curvature. But in these models, there is not only no singularity, but no event horizon and no black hole at all.
In short, there are no conditions that I am aware of under which the collapsing matter would fail to produce a singularity, but there would still be an event horizon and a black hole, with a singularity in the vacuum region.
Ibix said:
I think I'm just observing in more detail that this isn't a satisfactory model of gravitational collapse.
Again, I'm not sure what you mean. Of course the presence of singularities in any GR model is taken by most physicists to mean that GR breaks down at sufficiently large spacetime curvatures. But that doesn't seem to be what you're referring to here.