> This experiments seems to make no attempt to eliminate
  > the possibility that this is completely random.

Psychophysicists use two-alternate forced-choice experiments precisely to eliminate the possibility of the result being "completely random". Indeed, for this experimental design it---given sufficient trials---is quite easy to do hypothesis testing and compute confidence intervals for the hypothesis that the subjects can perform the task better than chance.

You are correct, of course, that this doesn't prove that they see the photons. Maybe the experimental apparatus was poorly made and accidentally a noise when the photons were flashed. Or maybe someone cheated and told them which trials the photons would be delivered. Or perhaps they simply have ESP. But the one thing that we can easily have confidence about is when an effect differs from random chance.