> Hawking radiation is due to process outside the black hole.

Hm, yes, the Hawking quanta originate outside the event horizon, and not very nearby outside either (Giddings 2015 <https://arxiv.org/abs/1511.08221>, Unruh "Dumb Holes" 2007 <http://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=43>).

However it's best to think of "the black hole" as the entire spacetime (in Hawking's 1974 treatment and similar; or alternatively out to somewhere in the asymptotic flatness), in which there are two regions without a horizon, one to the past of the event horizon formation, and one to the future of final evaporation.

What goes into the horizon doesn't stay in, therefore what happens inside is part of the picture (and has been speculated about for fifty years! Fifty!)

Yeah technically, in the current formulation, but I think at this point the smart money is on Hawking radiation being correlated with something on the inside. For instance this is my favorite solution for the information-loss problem, that the info is carried away by hawking radiation.