@knzhou is right in his comment. Light cannot escape from a black hole (BH) because gravity causes a high enough curvature that its paths (lightlike geodesics) outwards all become tangential to the horizon at the Horizon. They don't have to interact with anything, shooting them as straight out from inside the horizon as possible they simply cannot overcome the gravitational field. For particles with mass it is even worse, and the gravitational field keeps them in.
For the cosmological period before recombination the reason light has a hard time propagating is because the charged particles like protons and electrons present have such a high density that the photons mean free path is small enough that they can't get far before interacting or colliding with the charged particles, and further, any atoms that form get ionized through collisions quickly. It's a statistical effect due to the high density and high temperature. As the universe expands the density is lower, and so is the temperature, and the mean free path is longer, and they interact or collide less with electrons and protons, or any atoms formed, giving those charged particles a chance to collide more often and form hydrogen atoms. You can figure out at which temperature recombination will happen from various parameters and the staristical equilibrium equations (Botzman, etc) to calculate the equilibrium densities for both charged particles and uncharged hydrogen atoms, in the expanding universe (so you also need to use the expansion rate). The temperature at which many more hydrogen atoms can stay together comes out to be around 3100K (actually it happens over a range of maybe a few hundred degrees K). After that the density of charged particles is less, and photons have a much higher mean free path.
So the cosmological recombination effect is when the collisions which are keeping the charges free and ionizing any atoms that form slow down, the light collides less and is freer to drift out of the area. It has nothing to do with black holes, and it is well understood. Same thing happened when neutrino collisions with particles slowed down and neutrinos were set free. This happened at a higher temperature, and thus earlier, than photons.
Two different physical effects, those epochs when certain collisions slowed down and certain particles radiated out, and Black Holes. By the way, this recombination epoch was way way after the Planck epoch which lasted much less that 1 sec. Recombination was at about 380,000 years after the Big Bang. No need for quantum gravity at the recombination times.
