I am not talking about any particular observers. I am talking about invariant properties of the spacetime geometry and invariant curves within that geometry. Nothing I have said depends on any particular choice of observers. I only phrased it in terms of what a particular observer sees because the post I was responding to was asking about what can be seen from outside the horizon. I could just as easily have phrased it in terms of what null curves intersect with a particular timelike curve; the underlying geometric facts, and the fact that they are invariant, are the same either way.

> There is no global reference frame that allows you to say that the objects actually fell in and hit the singularity inside the hole long before you see the final image

Yes, there is. There are a number of frames that cover both the exterior and interior regions; for example, Painleve coordinates, Eddington-Finkelstein coordinates, and Kruskal coordinates.