They do speak of the gravitational singularity (not the event horizon). It's within the horizon, there is a contradiction between the need for two infallers' position to reach the singularity (thus hitting a point) and the causal impossibility of them actually meeting (following General Relativity?). This suggests that the point should maybe be a surface.

I don't follow most of the arguments however.

No, they are theorizing about the Kerr metric. The Kerr metric describes rotating black holes and instead of a point like singularity predicts a ring of zero thickness with infinite density additional work posits the ring is unstable and collapses to a surface all inside the event horizon, they then proceed to make predictions about the nature and behavior of that surface.

By the way the Kerr metric predicts a ring because the centrifugal acceleration due to the rotation partially counteracts the gravity. As far as I understand, not a physicist.