Also, I'm not sure you need stuff to always approach without ever getting there: it's not clear what happens to quantum fields at a singularity, but if you no longer have time displacement symmetry you don't get energy conservation so stuff might just cease to exist.
If you try to naively apply the same procedure you would use to define volumes outside the event horizon, it's proportional to the lifetime of the black hole (infinite in the classical case) for the reason you note.
Of course the better way to interpret this is probably just that the interior volume of a black hole is ill-defined.
Every geodesic that crosses the horizon terminates at the singularity in a finite amount of time. If you fall past the horizon the singularity is in your definite future and in a finite amount of proper observer time your geodesic will end, at which point ... who knows what happens.
The idea that no information can escape from the event horizon is model-dependent; it's a mathematical result from the assumptions of classic general relativity. If you operate with the assumption that the claim is true, it's fair to consider the other mathematical results of classic general relativity. That's why (some) people say there is a singularity.
Of course, the point is that if GR isn't exactly but only approximately true to a deeper theory, is the claim about singularities also true in the deeper theory, or does the deeper theory resolve the singularities somehow? That deeper theory may also say that information escapes from the event horizon.
For example, Hawking showed that semiclassical quantum corrections to GR means that black holes emit thermal (information-free) radiation. If we had a fully quantum theory of gravity people expect the evolution to be completely unitary and the information could ultimately be acquired (perhaps at enormous computational expense).