There is a whole light cone, which, in non-degenerated coordinates, looks like a normal, non-degenerated light cone.

That you cannot use the Schwarzschild metric itself at the horizon, because it is degenerated there, is well-known. You can argue that I should have guessed that he means coordinate speed in Schwarzschild coordinates, and, therefore, should have objected in a different way, namely that these coordinates are invalid at the horizon. But I have interpreted the original claim without making this hypothesis about the particular coordinates. There are other, non-degenerated choices of coordinates near the horizon, say, with a time coordinate which becomes proper time for some infalling observer. And, whatever non-degenerate coordinates you use, you will never have a light cone degenerated into a line, which would correspond to zero light speed in all directions.