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?
Not true. You incorrectly assume that when the star collapses there is already an event horizon, somewhere. But it isn't. The event horizon will only form at some particular proper time, and at that time most (but not all) of the stellar mass will be trapped by the horizon and will form a singularity (interestingly all the trapped matter will reach the singularity at the same proper time).
This is all very clearly explained here: http://www.aei.mpg.de/~rezzolla/lnotes/mondragone/collapse.p...
Related physics stack-exchange answer: http://physics.stackexchange.com/a/28837
I don't want to sound rude, but you really need to understand GR. The seminal work on the subject is "The Mathematical Theory of Black Holes" by Subrahmanyan Chandrasekhar, you need to know GR very well to be able to understand it, but the pdf I linked is comparatively much simpler on the GR and the mathematics required.
I made no such assumption. You don't need an event horizon to dilate time, just a massive gravitational field, and something resisting it.
When the sun collapses either the outer layers experience a repulsive force of some kind from below (a bounce perhaps, which is already theorized for supernovas, or maybe neutron degeneracy), or the outer layers accelerate to nearly the speed of light, either way they dilate time, and from the POV of an outside observer take an infinite amount of time to collapse.
Because of this dilation the rate of light output (as seen from outside) will drop and the sun will "turn off" - i.e. it will look [almost] identical to a black hole, except without the physics breaking singularity or event horizon.
One visible effect will be extreme red-shift of the light, which will look (to us) like a very distant object hubble-red-shifted by distance/time.
