Four layer PCB stackup with thick prepreg layer: how is it useful?
So, what am I missing? From this point of view, the thick-prepreg
  stackups of some manufacturers are useless. But they exist (and in
  fact are standard!) for some reason. How do people use them?


I think what you're  missing is the fact that majority of four-layer boards will not have ANY controlled impedance lines. Therefore, any old prepreg thickness will do.

Other than controlling trace impedances, four layer boards are useful for improving EMI performance by providing better grounding, allowing more dense component placement and simplifying routing.

It would be interesting to know what factors determine the PCB stackup in these cases, though. Maybe it is the ease of manufacture or cost of material?