[AlecBG]

well he certainly mentions a result where if there is an everywhere timelike Killing vector field (+ some other assumptions) you can prove that Hawking radiation doesn't occur and that does not include for example the Schwarzschild solution because the Killing vector field partial/partial t becomes non-timelike on the horizon.

So for example if you take a dead star in a vacuum with nothing else in the universe (and make certain technical assumptions) then you can prove that the star does not emit Hawking radiation. That's quite a strong result, and certainly does make the result seem shocking.