I am either badly misunderstanding what is being claimed here or what is being claimed here has got to simply be wrong, and this can be demonstrated with an extremely simple test.
Two quick scenes captured from my desk chair, each done at 500mm at f5.6 and at 200mm at f2.8. Shutter speed and ISO are kept the same across all photos. (Ignore the exposure bias of +.7: this was just there from the last time I was shooting outdoors, but for this test I just went to full manual and picked an ISO of 3200 and stuck with it). The 200mm shots were then cropped into be more or less the same size. I first did the water bottle, but then quickly thought maybe I should do something white because I planned on measuring the luminance of the target and thought it may be less finicky to pick an objective point to do this with off of a more uniform color and so I shot the UPC codes.
I ended up not taking the time to measure the luminance because very, very clearly there is more light on the targets with the 2.8 apertures and there is no need of looking at any numbers to prove this.
Now the entrance pupils here are about the same size as those given in several of the examples cited above: the 500mm at 5.6 is about 89mm, while the 200mm at 2.8 is about 71mm. As I understand the claim above, this should mean that there is more light on the subject with the 500mm lens and it would only be by looking at the full frame that we would see the same amount of light from the f2.8 lens. Yet very, very clearly this is not the case.