I don't think you comprehend your own link.

Hes not talking about density in its usual sense. The usage of quotes on "size" when he talks about the schwarzchild radius and then quotes again on "density" are interesting because it implies there is more to what he's saying than the literal/mathematical definition of those terms.

His definition of the volume of a black hole is broadly based on this schwarzchild radius, which depends on the mass of thr black hole in question, which is apparently essentially the limit at which you can escape the black hole. This implies this is an indirect measurement of force or velocity (i.e. "escape velocity" in orbital mechanics).

He also concludes with "might as well be." If the distribution of matter within the sphere formed by the schwarzchild radius is irrelevant, why that particular choice of words?

This all implies one thing: that black holes are particularly not well understood.

I find your link very interesting, but it doesnt help prove your point. What is "quantum gravity"? Does it tie gravity into the other 3 fundamental forces? How does it relate to black holes? Is the author saying that a black hole exists in both an infintesimal volume (thus leading to infinite density) and has the "size" of a schwarzchild radius (leading to a density defined by the equation he uses) simultaneously? If you can't tie down the "size", how do you tie down the volume, and by extension the density? It still seems density in black holes is something that is not well understood, if it in fact physically exists as we understand density in other mediums (i.e. atmospheric density).