Consider Alice approaching a black hole while Bob observes her journey from a distance. Due to the large gravitational field surrounding the black hole, Bob observes Alice her clock to slow down as she approaches the event horizon. Alice, on the other hand, sees Bob's clock speeding up as she approaches the horizon.
Once Alice reaches the horizon, her clock stops according to Bob's observation, while she notices no change of rate on her clock (proper time). On the other hand, when she looks at Bob, she observes time progressing at an incredibly high rate on his clock.
Now, in a heat death scenario for our universe, the black hole is bound to evaporate by Hawking radiation eventually. From Alice her perspective, 'eventually' might be very soon on her world line. I would therefore hypothesize that she will witness the evaporation just as she remains about to cross the event horizon. I would imagine the black hole to shrink before her very eyes; leaving Alice in a distant future where the black hole has evaporated.
By that time however, Alice would have been bombarded with a large, and probably lethal, dose of Hawking radiation. Is my analysis of the situation correct? If so, why are we told a different story by physicists? If not, where is the error in my reasoning? For instance, could it be that Alice does not observe time to progress infinitely fast in the rest of the universe, if she were to cross the event horizon?