I just realized I was replying to the wrong parent comment! So sorry. I was talking about a PBS Space Time episode on YouTube I saw in (or came across via) a different comment.
What they mean when saying that "time and space switch roles inside a black hole" is that in standard Schwarzschild coordinates, the direction given by the time coordinate becomes spacelike at the event horizon and, likewise, the radial coordinate becomes timelike. This statement is specific to Schwarzschild coordinates, though, or, more generally, any coordinate system that is singular/ill-defined at the event horizon. There are coordinate systems, however, that don't exhibit this pathological behavior at the horizon and, there, no switching occurs.