What do I see while falling into a black hole?

[PeterDonis]

It seems highly doubtful to me that he's just plain wrong

Saying "time and distance swap roles" is not so much wrong as misleading, because the words "time" and "distance" in that statement do not refer to what you are intuitively thinking they refer to. They refer to coordinates, but you are intuitively thinking of them as the "time" and "distance" you experience in everyday life. The professor might simply not realize how misleading the statement is, because he understands that he's only talking about coordinates, so he does not draw the mistaken inferences from his statements that you are drawing. If so, he would not be alone; I have seen similar statements in many discussions of black holes, including some textbooks (not GR textbooks, but textbooks on other things that happen to mention black holes).

 

[Nugatory]

So I'd love to know what the problem is. But if you can't spare 10 minutes, perhaps someone else could.

There's a pretty good explanation here: https://arxiv.org/abs/0804.3619
The math in this paper is well beyond what belongs in a B-level thread, but Krasnikov's qualitative explanations of what it means may still be helpful. The bit about time and space switching or not switching roles is at the "remark" following equation 6; all Krasnikov needs to do is not use the same two symbols ($r$ and $t$) to label different things and the confusion disappears.