> Since this is true even for infalling observers getting arbitrarily close...

Arbitrarily close *hovering* observers will never see anything cross the horizon because of the extreme redshift, but *infalling* observers are different: they *fall in*. Proper time depends on the world-line, and different observers follow different world-lines, so "never" depends on which world-line we're talking about. The hovering observer never sees the infalling observer cross the horizon, but the infalling observer does.

Consider the Penrose diagram for a non-rotating black hole formed by a collapsing star in the context of classical general relativity:

[![enter image description here][1]][1]

In this diagram, diagonal lines represent lightlike directions. (Beware that this diagram distorts lengths and time intervals in order to make all diagonal lines correspond to lightlike directions. Representing a 4-d curved spacetime on a 2-d flat space requires making compromises!) The diagonal dashed line is the event horizon, and the horizontal line at the top is the central signularity. The diagonal lines labelled $\infty$ are in the infinite past and future, respectively. (Technically, these represent *lightlike* past and future infinity. The point where they meet represents *spacelike* infinity.) The lines labelled $r=0$ represent the center of spherical symmetry: each point in the interior of the Penrose diagram represents a whole sphere centered on $r=0$. Dashed lines can be crossed; solid lines cannot.

Now, consider the diagrams below, each of which shows the top part of the preceding Penrose diagram: 

[![enter image description here][2]][2]

- In the diagram on the left, the red line shows an infalling observer $A$ that hits the "surface" of the star just *before* it crosses the event horizon. The blue line represents light emanating from the collision event. The collision of $A$ with the star can be observed from the outside, albeit extremely redshifted (even though I drew it as a blue line), possibly into galactic-scale wavelengths if the collision occurs late enough. 

- In the middle diagram, the red line shows an infalling observer $B$ that hits the "surface" of the star *after* it crosses the event horizon. The blue line represents light emanating from the collision event. That light never escapes; it ends up hitting the singularity instead. For that reason, the collision of $B$ with the star *cannot* be observed from the outside: nothing that has crossed the event horizon can be observed from the outside. 

- The diagram on the right shows an observer $C$ that races inward but can't catch up with the collapsing star, because that observer's inward journey began too late. 

> we will always see and hit the original star at a point where it is still larger than its Schwarzschild radius?

Always? No, not always. We *can* do that, if we start the journey soon enough (case $A$), but if we start later, then we won't be able to catch up with the star until after crossing the horizon (case $B$), and if we start *too* late, we won't be able catch up to it at all (case $C$).


  [1]: https://i.sstatic.net/rn6bO.png
  [2]: https://i.sstatic.net/PfwnU.png