> 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

I'm not sure what you mean by "photons from the past". You can't "move in space" inside the horizon in any way that's fundamentally different from what you can do outside the horizon. And any photons you encounter will, by definition, come from somewhere in the past light cone of the event where you encounter them; that's equally true inside and outside the horizon.

> 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

No. The singularity itself is not a place in space; it's a moment of time, which is in the future of every event inside the horizon. That is the best, coordinate-independent way of describing what happens inside the horizon: inside the horizon, your future is now limited in duration by the presence of the singularity, a moment of time that marks the end of your future. There is no such moment of time outside the horizon. It has nothing to do with "space" or "space and time switching" at all. It's simply a future limitation on time.

> the mathematical explanation of sign flipping

Is only true in one particular set of coordinates. As such, it cannot describe anything physically real. That is a general rule in GR: if it's something you can make go away by switching coordinates, it isn't physically real.

> at the instant you cross the event horizon, whatever metric you're using is going to become degenerate

No. The metric used to draw the diagrams around minute 7 ish in the video is not degenerate anywhere, including on the horizon.

The coordinate-independent, physical thing that picks out the horizon is that radially outgoing light rays have zero expansion--that is, a 2-sphere of radially outgoing light rays at the horizon does not increase in area. In fact, the horizon can be said to be a 2-sphere of radially outgoing light rays with constant area.

> 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

This isn't a property of a "dimension", it's a property of particular curves in spacetime that observers can follow: such curves are called "timelike" to acknowledge this property. The overall property of spacetime that is related to it is that which direction along timelike curves is the "future" direction must be continuous everywhere--i.e., if you continuously move from one timelike curve to another, the "future" direction stays the same everywhere, it never switches.

> 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.

No, it doesn't. Spacelike and timelike vectors and their properties are the same inside the horizon as outside, and the direction towards the singularity that has the property you describe is still the timelike one.

It is true that, at any event inside the horizon, there will be spacelike as well as timelike curves that connect that event to the singularity, but that is also true outside the horizon for any moment of time to the future of a given event (since a "moment of time" outside the horizon is a spacelike 3-surface extending through the entire universe, and there will be points on any such 3-surface that are spacelike separated from any chosen event to the past of the surface).

> I'm sure this all goes out the window in case of a rotating black hole

Some of it does, but some of it is still applicable. You are correct that the singularity in Kerr spacetime is a ring, and it is also timelike, not spacelike, so it can be correctly described as a ring "in space" that "stays at the same place" as time progresses. However, the ring singularity is inside an inner horizon, and it is impossible to reach the ring singularity without passing through the inner horizon; and from the viewpoint of an observer between the horizons (inside the outer horizon/event horizon, and outside the inner horizon), the inner horizon has some of the same properties as the singularity in Schwarzschild spacetime does, including the key property that it's impossible to avoid it once you're inside the outer horizon--it's unavoidably in your future.

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

I think you are illustrating why I said the video is highly misleading. You are taking some reasonably valid intuitions from differential topology, but combining them with things the video is saying is leading you to incorrect conclusions.