Its an interactive attack, relying on "amplifying" side channel effects
adaptively. Thats like being able to poke the bird and make it tweet (in Jonathan's original analogy). This is quite different from sampling a signal from a system non-interactively. If you get to ask questions you can bias answers. So triggering the "bird" to tweet relative to the low bandwidth sampling boundary allows us to probe data within the structure of a signal we could not possibly sample under Nyquist restrictions. But a public-facing server would regularly be "tweeting", no?