The article is a bit dodgy in this regard. It just means that at a quantum level, the energy fluctuations (as observed in this experiment) are not big enough to cause significant space-time alterations.
So, it is foamy, but not as foamy as other people thought (the idea of virtual particles having no gravitational contribution sits uncomfortably in my mind).
The average would be unchanged, but the whole point of their measurement is to measure the resulting statistical broadening of the event.
That is, if the burst lasted 1 second (I made this number up!), you'd expect all photons to arrive within 1 second of each other. But if space-time foam had a strong effect, you would expect the same average time, but you might expect the spread of the data to be larger: maybe 5 seconds (also made up). (Essentially, randomness like this would be expected to increase the standard deviation of the arrival times.)
So by measuring the spread of the photon arrival times, you can put an upper bound on how big the space-time foam effects can be: the actual spread is (very roughly) the sum of the actual length of the event plus the spreading due to space-time foam. These folks are claiming that for certain models of space-time foam, the photons they observed arrived too close together to be consistent with the existence of that foam at the expected scale (assuming it's not a statistical fluke).
If the situation was chaotic (which I would presume a system with many random perturbations to be) the expectation is for the repeated small changes to have a significant impact.
My guess would be this would be true only if they were coherent, and the resonant frequency were some multiple of the Planck length. The many small random perturbations would largely cancel each other out and the remainder would be insignificant (not energetic enough) to meet the lowest energy requirements to nudge a photon by even a miniscule amount.