The first sentence does not imply the second sentence. This answer is wrong, specifically the second sentence is wrong. I am not certain that the first sentence is correct either, although it might be.

This question clearly is about Classical GR. The OP did not mention the evaporation or Hawking radiation from unproven semiclassical gravity. Regardless this answer is incomplete, because it describes a semiclassical case only from the standpoint of Classical GR. The semiclassical view of the same case is that the falling observer evaporates before he reaches the horizon. The irreconcilable difference in these two viewpoints is called the Information Paradox. If the Hawking radiation exists, then GR breaks near the horizon where matter is destroyed by the firewall singularity.

@safesphere Classical GR is not applicable to the timespans, comparable to the BH evaporation time. From the OP question it seems, he is interested in what will happen in reality.

The Hawking radiation (1) is not predicted theoretically by a confirmed theory; (2) is not experimentally observed; (3) violates fundamental laws of physics via the Information and Page Time Paradoxes. Hence, as far as science is concerned, the Hawking radiation does not exist, but is a minor theoretical artifact of merging by force two incompatible theories, classical and quantum. The current confirmed theory of gravity, regardless of the timeframe, is Classical GR. Plus, es explained above, your answer doesn’t consistently follow semiclassical gravity anyway.

@safesphere something reaching under the event horizon and the very black hole forming also violates the information paradox. As to the firewall, I did not say it does not exist. It is evident that if the BH explodes ahead, you are a subject of intense radiation that evaporates you. So, I believe firewall exists but it is out of the scope of this question.

It is not controversial that someone falling into a black hole will do so in a very short time, according to his own clocks, regardless of what distant observers observe. Just like the relativity of simultaneity in SR, the concept of "now" is different for the falling person and distantobservers.

@D.Halsey their short time corresponds to infinite time by the clock of the distant observer, which exceeds the time of the black hole evaporation. You are using pure GR outside its domain of applicability. The remote observer will allways see the falling observer outside the BH even after the BH ceases to exist. So, he will be able to meet with the falling observer again after the BH is no more (if he is not destroyed by the explosion).

@Anixx No, you are comparing incompatible notions of time. Black hole evaporation happens in the time close to the black hole.

@D.Halsey I do not understand what you wrote.

@Anixx Everything that's ever fallen to a black hole (including the original star matter) remains just outside the horizon forever, as observed from outside. Imagine one of these objects fallen a while ago is an observer named Bob. Next we wait for a gazillion years until the black hole has evaporated. This means that everything that has accumulated at the horizon is gone, including Bob. As far as we are concerned, Bob has evaporated along with the rest of the matter. And yet Bob himself expects to survive exactly the way you describe. This conflict of views represents the Information Paradox.

@safesphere "forever" means also after the BH is no more. Hardly Bob can survive if he is close enough to the BH though by the time the BH explodes. Yet, being destroyed by an explosion does not mean the information gets lost.

@Anixx You still don't seem to get it. When the BH evaporates, what exactly evaporates? The matter the BH consists of. This matter is outside the horizon (classically forever, but in this case until it has evaporated). Bob is not special. He is just a speck in the mass of this matter. So he evaporates too, because, if he doesn't, then nothing else does either - he is not special in any way. And so he evaporates in the view of a remote observer, but not in his own view, as you describe. There are two conflicting views in semiclassical gravity. You are describing only one and missing the other.

@D.Halsey Your objections are incorrect. There is no discrepancy in calculating the time of the final evaporation in the coordinates of a free falling observer vs. a static remote observer. You are confusing simultaneity with causality. Simultaneity is about two events happening at a distance. In this case however, there is only one event, the final evaporation, so the issue of simultaneity does not arise. Anixx is correct with his reference to the time periods.

@safesphere oh if you are at this, I should point that the very formation of the event horizon in finite time is impossible. So, if we do not just assume that a BH "just exists" with certain mass (that never fell in), then the picture is that the horizon never forms. No BH can be fully formed in finite time.

@Anixx This is correct on the cosmological timeline. However they say that a BH causes a turn in the direction of time by $45^o$ at the horizon (cosmological eternity) and then passed $45^o$ to cosmologically "imaginary time", meaning in the formerly spacelike radial direction, This means they extend spacetime beyond the cosmological eternity at the horizon. While this looks "cool" (and brings them a lot of research grants and even Nobel Prizes), there are a couple issues with this. (1) This extension is not a fact, but an unproven assumption; (2) playing with infinities is a dangerous game.