Black Hole Puzzle

Not quite. He will see the light emitted by _all_ of the matter that has fallen in before him, but only in an infinitely small area.

Penrose diagrams draw black holes as if they have existed forever, and will last forever -- that's what the "future infinity" line means.

Obviously black holes form at some finite time, and Stephen Hawking showed that they evaporate in a finite time.

This matters. A lot!

Fundamentally, relativity does not allow observers to disagree on observations of what, only when. If an outside observer never observes someone falling into a black hole then they cannot fall in. It's just that simple! Again, for the slow people at the back of the class: Observers cannot disagree on this. If there is a paradox in your model, then your model is broken, end of story.

There is this weird aspect of modern physics of holding on to the almost-mystical "woo" aspects far more than is justified in either the mathematics or observations. Quantum Mechanics and cosmology are especially rife with popularisations of what amounts to science fiction story telling. It's fun to think about wormholes, white holes, alternate universe accessible trough black holes, etc... I've read these novels, they were fun! Not physical though, natch.

Much more realistically: An infalling observer is slowed down relative to the outside universe and the stellar remnant that formed the black hole. Effectively, the hapless infalling victim sees the black hole and the universe both "speed up".

The error most people make here is that they assume that the black hole is already fully formed in the infinite past and is "completed", making it a mathematically perfect sphere in a sense.

No!

It hasn't finished forming because as its gravitational field increases its time distortion increases. Its formation is "frozen in time" (actually just very very slow), it is never fully formed.

Infalling observers see this slowdown speeding back up, so what they observe is the light of the black hole evaporation getting blue-shifted and brighter until right before mathematically they would have "fallen past the event horizon" what they're actually seeing is akin to a supernova exploding in their face. They're blasted into subatomic particles, joining the radiation of the black hole evaporation many trillions of years into the future... but not infinitely into the future.

This model solves every paradox of black holes, making them boring and not sci-fi exciting again, which is why you haven't heard about it! Uncool stories don't get repeated on blogs for ad-clicks.

Just remember: there are no infinities in the universe, and anyone using one in their theory is almost certainly just making stuff up to sound cool.

PS: Penrose is also behind the kook notion that brains are quantum despite all evidence.

This universe is bizarre af, it shouldn’t even exist and us too but somehow it does.

In the classical approach, the light that the object emits a second before crossing the horizon will take years to reach the observer, at a millisecond it'll take decades, at a microsecond it'll take centuries, etc. But it's always still coming, never stops completely. So in the classical analysis, an observer will always see the object, unboundedly redshifted, infinitely near the horizon, but not quite there, forever.

However, in the presence of Hawking radiation, the black hole loses mass, and thus the horizon shrinks. When this happens, that last bit of light the object emitted before crossing the horizon will now reach the observer in finite time. In particular, the observer will see that long _before_ they see the black hole evaporate entirely; one photon's worth of radiation is enough to do the trick.

You're correct that the object will experience this time dilation by seeing more radiation as it gets closer to the horizon. It'll also start getting pelted from behind by infalling radiation. But nothing will blow up as it crosses the horizon, so it'll safely continue on to experience whatever quantum-graviational phenomena happens at/near the "singlularity".

Generally you can search for finite black holes and read things written in the last half a decade or so.