When all the particles above the horizon are escaping and all the particles below the horizon are let to have a velocity closer to c in the outward direction than the escaping particles have, GR still demands that the cloud splits, unlike what would be predicted for an inertial frame with the same velocities for particles specified. In an inertial frame under those conditions the particles with higher velocities would be moving closer to the other particles, not getting further away from them. There is a contradiction.

This is actually a better statement of the argument than the one given on the web page (better because it's more concise and doesn't get bogged down in irrelevancies); in fact it makes it clearer to me just what the argument is actually supposed to be. However, it's still not a valid argument against GR. Here's why: GR agrees that, with respect to Frame X, the particle above the horizon and the particle below the horizon get closer together! But only within the confines of Frame X, and only with respect to distance relative to Frame X.

Distance is frame-dependent; distance relative to Frame X is not the same thing as distance with respect to the radial coordinate r that is used in a global coordinate chart. The fact that the particles are getting closer with respect to distance relative to Frame X does not mean that they are getting closer together with respect to the coordinate r; in fact they are not. The particle inside the horizon has an r coordinate that decreases with time, while the one outside the horizon has an r coordinate that increases with time (here "time" is time with respect to Frame X).

Also, the fact that the particles are getting closer together with respect to Frame X does not mean that they will be getting closer together with respect to some other local inertial frame centered on an event further along the horizon. Local inertial frames centered on different events don't "line up" in curved spacetime the way they do in flat spacetime; that's one way of expressing what it means for a spacetime to be curved. So you can't assume that, just because the particles are getting closer together with respect to Frame X, they will continue to get closer together once they've left Frame X, much less that they will continue to get closer together until they meet. That deduction would be valid in flat spacetime, but it isn't in curved spacetime.

When you say that singularities have a location in time only you can't then refer to their location in space without contradicting yourself. The "in the center of the black hole there" references a place in space. You'd have to instead be careful to say something like "you won't be pointing to the singularity that you say is in the center of the black hole there".

Fine, consider my statement suitably modified (it should have been obvious from context that that's what I meant anyway, but if you insist on quibbling over language, so be it). As modified, the statement is true.

Instead of quibbling over language, you could have asked a relevant question: what are you pointing at when you point at the center of the galaxy? The problem with this question is that it assumes that there is a unique answer to it; that there is some unique "object" sitting "at the center of the galaxy". In an ordinary spacetime without a black hole present, that would be a valid assumption; but it isn't when there is a black hole present. In one sense, when you point at the center of the galaxy, you could be said to be pointing at the object or objects that originally collapsed to form the black hole there; in another sense, you could be said to be pointing at the hole's horizon. Once again, a spacetime with a black hole in it doesn't work like the ordinary flat spacetime you are used to, or even like an ordinary curved spacetime with a planet or star in the center instead of a black hole (so that there is no event horizon and no singularity).

I first saw this blog yesterday.

But haven't scenarios at least something like this come up in your classes?