I'll even quote:
"At some moment, the radius of the star drops below the Schwarzschild radius - that's where the yellow and orange lines intersect. At this moment, the star is pretty much doomed. Right now I am not sure whether one can generally prove that the event horizon becomes inevitable at this point but I am sure that it is going to be there in a moment according to the picture above which is surely realistic."
He simply skips right past the impossible point: "the radius of the star drops below the Schwarzschild radius". It will take that Sun an infinite amount of time to do that. Any conclusions you draw after that point are unscientific.
He draws all sort of conclusions from the diagram, without ever showing that his diagram is actually true.
There was some talk of how to be a stationary observer near the black hole you must keep accelerating, but a: I fail to see how that explains the infinite time, and b: who said I have to be near the black hole to look at it?