I don't understand the essay at all, perhaps because I'm not a domain expert. Edges are clearly a very strong signal in the visual system -- as the article points out, edge detection is one of the first things that happens in the visual pathway from the eyes. However, edges are clearly not the
only signal, as the article demonstrates with cross-hatching, colour, and so on (and is also obvious to presumably all sighted humans).
If you remove edges as a signal -- say, by taking the coloured apple in the article, removing the lines, and blurring the colours around the silhouette -- you'd probably still recognise an apple, but not as quickly. For the same reason, if you defocus your eyes (or take off your glasses -- the popularity of these is a strong indicator that contour matters), you'll have more difficulty navigating, even though many signals (colour, shade, depth) are still present. Clearly edges are important, but are not the only thing we're working with.
Optical illusions also don't invalidate the hypothesis, because they almost by definition rarely occur in nature. Similarly, objects in extreme shadow don't invalidate it either, because, e.g., we are quite capable of recognising half a lion as a lion.
I think possibly there's a difference in interpretation here. The claim the author has an issue with is (his phrasing): "the lines in a line drawing are drawn at natural image edges, where an edge receptor would fire. These lines activate the same edge receptor cells that the natural image would. Hence, the line drawing produces a cortical response that is very similar to that of some natural image, and thus you perceive the drawing and the photograph in roughly the same way."
But the paper the author links to doesn't say that. It says "The likely explanation is that lines trigger a neural response that has evolved to deal with natural scenes." It's not claiming that line drawings and photographs are perceived in "roughly the same way", only that we evolved to recognise outlines and line drawings are outlines or at least contain outlines.
So problem 1 (what about all the other features), problem 3 (what is the benefit), problem 4 (visual art isn't just line drawings), and problem 5 (edge detection is not a line drawing algorithm) don't seem to really address the hypothesis.
That just leaves problem 2, "we can't see internal representations". I'm not sure what the statement is here. The author writes "The idea is that we have neurons that activate for object contours and similar, and that line drawings directly activate these neurons. Lines-As-Edges is a special case of this hypothesis. I don’t understand this claim at all." But that claim, as a hypothesis, seems very reasonable? The author seems to be saying that you can't bypass all the neuronal machinery to get directly to the contour-recognising neurons. But that's not true if all you're bypassing is the stuff to isolate the feature of interest (edges, in this case).