The Schwarzschild black hole is one of very few analytically tractable black hole configurations in GR. It is a non-spinning, non-charged, spherically symmetric, isolated, and static black hole. The instant we start throwing mass into it, several of those idealizations break, so we can’t necessarily expect to use Schwarzschild solutions to understand the growth of a black hole.
That’s the long and short of it. For something more to hang your hat on, imagine not throwing a rock into the black hole but rather a uniform spherically symmetric shell of dust concentric with the black hole. The static Schwarzschild solution would say that the dust never reaches the horizon at radius R_S, but considering the mass of the black hole plus the mass of the dust shell, there is some larger radius R_S' that will eventually be the new event horizon, and the dust will reach R_S' in finite time if we approximate the black hole as only having its original mass. So, at some point during the system’s evolution, we definitely need to update our geometry.
This isn’t a path to actually solving the system. It’s just to show that more care is clearly needed. In situations where significant mass is infalling (say, with binary star mergers), one goes immediately to numerical simulations to calculate what remote observers would see.
Yeah, I find it a little misleading to say that the escape velocity exceeds c, since there is no such velocity. Recalling the recent speed of light thread, space and time are geometrically related, and a speed greater than c is as irrelevant as “north of the north pole”. There are common diagrammatic representations of spacetime around a black hole where you could draw a line that looks like a trajectory of something going faster than c and subsequently exiting the horizon, but that’s not an explanation per se, even if it’s also not accidental.
A geometric way to state the situation is that, when caught by a black hole, your continued progress through spacetime always decreases your distance from the center.
Edited to add:
@Chronos’s version stating:
is valid, but keep in mind that the timelike-ness doesn’t have to be locally obvious. This is where I would give pause to the “space and time switch” story. It’s true that, some time in the future, you will definitely reach the singularity. But right now, time and space can look quite normal to you. The infalling observer’s watch doesn’t suddenly start measuring distance nor does his ruler measure time. But the curvature of spacetime is such that all trajectories lead inevitably to one place.