I agree that for a spacetime that is exactly Schwarzschild, the infalling observer does not see the entire history of the universe. However, this turns out not to be the generic case you would expect for an astrophysical black hole, which formed from collapse of some approximately spherical distribution of matter. This topic is actually being actively researched, and there are some very interesting results about what the inside of a black hole actually looks like. See, for example, this recent paper.

The reason that in Schwarzschild the infalling observer does not see the entire history of the universe is that the singularity is spacelike. This means that there is a range of points where the infalling observer can hit the singularity, and each point can only see part of the universe in its causal past.

But people have known of other kinds of black holes for a long time that do not share this behavior. The best know examples are the Reissner-Nordstrom solution for a charged, spherically symmetric black hole, and the Kerr solution for a spinning black hole. These both have timelike singularities, and hence the situation is quite different. Here is a causal diagram of a Reissner-Norstrom black hole:

The vertical jagged lines represent the timelike singularities for this black hole. In this case it is possible to avoid the horizon and emerge into a new universe that you could attach to the top of this picture. In this case, when you cross the inner horizon, you should be able to look back and see the entire history or the universe.

This brings up a problematic point however. The observer passes the inner horizon in finite proper time, yet he is able to see all the light that enters the black hole from the entire infinite history of the universe. Since light has energy, you might think that this pile up of radiation from the outside universe should lead to a great deal of curvature, and indeed it does. This is known as a mass inflation instability of the black hole. Kerr black holes share this feature, although the structure of the singularity in that case is more complicated.

So for generic black holes that are not exactly Schwarzschild, a different behavior is expected. The perturbations tend to change the singularity from being spacelike to behaving like a null surface, i.e. following the trajectories of light. A picture from the above paper shows this situation:

The outside universe lives in the bottom right triangle of this picture. The lines labeled $\mathcal{CH}^+$ are the null singularities. The paper found that this situation resulted from perturbing the Schwarzschild solution with scalar field matter. In this case if you fell into the black hole from the outside universe, you would run into the null singularities, and assuming you hit the one on the right, you will see everything the entire history of the universe, in the sense that all that you will have access to light that enters the black hole from arbitrarily late times of the universe's history.

I agree that for a spacetime that is exactly Schwarzschild, the infalling observer does not see the entire history of the universe. However, this turns out not to be the generic case you would expect for an astrophysical black hole, which formed from collapse of some approximately spherical distribution of matter. This topic is actually being actively researched, and there are some very interesting results about what the inside of a black hole actually looks like. See, for example, this recent paper.

The reason that in Schwarzschild the infalling observer does not see the entire history of the universe is that the singularity is spacelike. This means that there is a range of points where the infalling observer can hit the singularity, and each point can only see part of the universe in its causal past.

But people have known of other kinds of black holes for a long time that do not share this behavior. The best know examples are the Reissner-Nordstrom solution for a charged, spherically symmetric black hole, and the Kerr solution for a spinning black hole. These both have timelike singularities, and hence the situation is quite different. Here is a causal diagram of a Reissner-Norstrom black hole:

The vertical jagged lines represent the timelike singularities for this black hole. In this case it is possible to avoid the singularity and emerge into a new universe that you could attach to the top of this picture. In this case, when you cross the inner horizon, you should be able to look back and see the entire history or the universe.

This brings up a problematic point however. The observer passes the inner horizon in finite proper time, yet he is able to see all the light that enters the black hole from the entire infinite history of the universe. Since light has energy, you might think that this pile up of radiation from the outside universe should lead to a great deal of curvature, and indeed it does. This is known as a mass inflation instability of the black hole. Kerr black holes share this feature, although the structure of the singularity in that case is more complicated.

So for generic black holes that are not exactly Schwarzschild, a different behavior is expected. The perturbations tend to change the singularity from being spacelike to behaving like a null surface, i.e. following the trajectories of light. A picture from the above paper shows this situation:

The outside universe lives in the bottom right triangle of this picture. The lines labeled $\mathcal{CH}^+$ are the null singularities. The paper found that this situation resulted from perturbing the Schwarzschild solution with scalar field matter. In this case if you fell into the black hole from the outside universe, you would run into the null singularities, and assuming you hit the one on the right, you will see everything the entire history of the universe, in the sense that all that you will have access to light that enters the black hole from arbitrarily late times of the universe's history.