When real physical objects become black holes, the light they emit to distant observers becomes highly redshifted as the event horizon size is approached as seen by distant observers.

As the surface of the star reaches the horizon, its light is infinitely redshifted. In the reference frame of the star's surface, as it just passes the horizon radius, it has emitted a finite number of photons just outside of the horizon, and it is these that may escape to distant observers. The problem is that they are increasingly redshifted as the surface emitting them gets closer to the event horizon. In the limit, the last photon is almost infinitely redshifted. Also, gravitational time dilation occurs so that the interval between the emission of each photon lengthens dramatically as the event horizon is approached, and the intensity of the light exponentially decreases in time. The net effect is that to the outside observer, the surface of the star fades within a few seconds and its wavelength of peak emission shifts from X-ray to optical energies, all the way out to infrared and radio wavelengths. X-ray astronomers recently discovered just this fading effect in the X-ray light from a black hole a few thousand light years from Earth.

Now, the view is completely different from the reference frame of the star. Nothing weird happens when the surface crosses the event horizon radius. Even the view out into distant space may be unchanged except that incoming light get highly blueshifted as it falls down the gravitational well of the black hole and passes the horizon. But, after passage, no light can escape the hole, and furthermore, the mathematics show that no stable orbits are possible inside. Space-time itself is collapsing.

So, does a star look 'frozen' to outside observers? I do not see how this is avoided except that the frozen surface is so highly redshifted that you cannot see it by its emitted or reflected light. It becomes a perfectly black surface hovering millimeters above its event horizon for all time to come in OUR reference frame. So far as the collapsing star is concerned, however, it passes through the horizon and reaches the Singularity in a fraction of a second.

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