Stellar-mass black hole formation sequence

[PeterDonis]

You don't have to have a region of empty space per se; it can simply be a region of lower density.

I suggest taking some time to work out the math. It isn't as simple as you are assuming it is.

(As I note below, there is no known analytical solution to the differential equations for the case of nonzero pressure; but you can still look at the equations themselves and work out some qualitative features.)

Is there some aspect of the Oppenheimer-Snyder model which would prevent the core collapse from "outrunning" the collapse of the rest of the star?

Yes; but that aspect is that the O-S model assumes zero pressure, so it eliminates the only possible thing that could slow down the collapse of any part of the star.

AFAIK there is no analytical solution for the case of nonzero pressure, so the only way to study that case would be to do so numerically. I know such numerical simulations have been done, but unfortunately I'm not familiar enough with the numerical relativity literature to be able to point to specific results.