The problem is probably connecting an image of a tilted causal cone at p near but outside a regularized r_s in Schwarzschild spacetime with the idea that from p there is a limited number of null geodesics escaping to the asymptotically flat region, that the fraction decreases with decreasing regularized r coordinate, and that from p' at the trapping surface there are no such null geodesics, as all non-spacelike curves (accelerated or not) at any point on or interior to the trapping surface terminate at the singularity. The idea that the singularity inevitably lies in the future of
any observer at p' is behind the "spacetime swap" notion.
Some of the problem is that Schwarzschild coordinates have surprises buried in them, and what \Delta r and \Delta t mean are not what most people tend to think.
Someone should do an ELI12 of Unruh's (ca. 2014) excellent (give or take varying the spelling of Martin Kruskal's surname) Schwarzschild BH global coordinates pedagogic review http://theory.physics.ubc.ca/530-21/bh-coords2.pdf> and add in a bit on Fermi normal coordinates as a maybe-obvious not-a-chart follow-on to the commenary just above eqn (55). But on "maybe-obvious", Unruh has the choice line: "Since in a large number of cases, the single horizon coordinates were discovered long before Schild’s coordinates, this is an exercise in alternate reality – what could have so easily happened if only the generators of those coordinate systems had recognized what they had."