My understanding of the explanation and the video was that they don't literally switch roles, in the sense that you could then travel backward in time, or couldn't alter your position with respect to the singularity. It's more that, in terms of light cones, inside the event horizon, you can move in space in such a way encountering photons from the past, which is somewhat like travelling back in time. So, in a sense, the "arrow of time" doesn't work inside the black hole. But, since all world lines do eventually end at the singularity, which basically means that no matter what you do, you're always moving toward it, space becomes time-like in the sense that there's then an "arrow of space," analogous to the "arrow of time."

I found the mathematical explanation of sign flipping to be fairly easy to understand. One thing in particular that equation makes clear is that at the instant you cross the event horizon, whatever metric you're using is going to become degenerate. That makes some sense, because you're literally crossing over into a different realm there, but, it doesn't seem physical to me, unless it's not actually physically possible for matter to literally fall into a black hole; rather that matter we perceive as falling into it gets smeared out across the event horizon, its information being "painted on" to the event horizon. (This reminds me of the "black hole information paradox," but I'm not sure exactly how it all plays into it. For small enough holes and small enough particles, I would guess there is some interaction between the particle approaching the event horizon and Hawking radiation pushing back on it, kind of like solar wind does. But, that's pure speculation.)

Physically, the thing that made the most sense to me was that of the 4 spacetime dimensions, outside of a black hole, the one we have designated as "time" is so designated because it doesn't behave like the other 3, because we can only travel in one direction along it. Inside the event horizon, it's clear that one spatial dimension corresponding to the vector extending outward from the singularity to your position acquires this property.

Because they only did the example of a Schwartzchild black hole, I'm sure this all goes out the window in case of a rotating black hole, in which you no longer have a point singularity, but a ring singularity [0].

Since I have never specifically studied GR, all my actual black hole knowledge comes from intuition derived from a differential topology course. But, that intuition is telling me that something similar happens in this general case, except that space never actually becomes time-like. World lines do all still end at some point on the singularity, and you can still get weirdness encountering photons you might think you couldn't, so, my physics layman's intuition is telling me what you have is basically all 4 dimensions acquiring a similar character. That is, spatial dimensions start becoming somewhat time-like, and time starts becoming somewhat space-like.

Differential topology gave me a really crude understanding of what goes on here, but, I do understand that black holes are fundamentally simple objects. They only have three properties: mass, charge, and angular momentum; so, you should be able to describe them fully from any reference frame outside the event horizon using those 3 variables plus the spatiotemporal location of the hole.

At that point, if you have a universe with nothing in it but your one, lonely black hole, you have a static universe. If you introduce matter, you could theoretically model interactions between all the particles of matter in your toy universe, and the black hole. If you constrain the number of particles enough, you should be able to at least approximate how this idealized universe evolves.

Is this a fair interpretation, or am I way off base?