[tsimionescu]

If that were true, then black holes would appear as extremely bright balls of red-shifted radiation, as all particles whose trajectory ever moved towards the center of the black hole would still be visible. This is obviously not true, black holes appear as completely black objects that might have an extremely bright ring or halo of matter orbiting.

[thebricklayr]

For a region of space with time dilation approaching infinity, red shift to temperatures below even the microwave background would be nothing.