PeterDonis said:
The short answer is that the "mass" of a black hole is a global geometric property of the spacetime. It cannot be localized to a particular point or region in the spacetime. For a longer answer, see below.
using the following part of your input:
"Komar mass", which corresponds to the intuitive idea of adding up the contributions of all the little pieces of matter inside the object to get a total mass. while they are stationary. they contain no matter; and if we extend our model to include the matter that collapsed to form the hole, that matter is not stationary (because it collapsed), so it has no well-defined Komar mass. So the ADM mass is the only mass integral quantity that works for black holes.
However, this method cannot be used for black holes, because considered as isolated objects in themselves, while they are stationary, they contain no matter; and if we extend our model to include the matter that collapsed to form the hole, that matter is not stationary (because it collapsed), so it has no well-defined Komar mass.
For me the logical reasoning and your input above together with the behavior of quantum mechanics that mass is distributed would then lead to the following:
assuming mass to be able to to be contracted without limits we get the situation you describe with no "Komar mass" in the spacetime region and an ADM mass seen from outside the black hole.
But assuming mass that cannot contract without limits (when it has a statistical determined volume) and while the large star that forms the black hole star collapses, I expect the curvature would reach a certain maximum (being determined by the "Komar mass" inside a sphere with certain radius) and then reduces to 0 in the center where the radius is 0 and a given mass density . (similar to stars, white dwarf or neutron star) so as the curvature does not go to infinity, the mass stops contracting in the region where the curvature reduces from a maximum to 0, and a stable mass distribution (although still extreme density) remains. And then a stable situation would still give a valid "Komar mass".
so based on that I conclude we could have 2 ways to describe the internal structure of the black hole, depending om the assumption that mass can contract infinite or not.
so then again I come back to the question what makes us so sure that mass can contract infinite under gravity.