The main issue I am skeptical of is the statistics rather than the neuroscience. Just because the brain can be stored in 10^10 bits does not imply that measuring O(10^10) bits at random will give you what you want. But perhaps Paul has a reason to believe this beyond e.g. the intuition from the fact that random projections work for compressed sensing (which seems qualitatively different to me, since recovering L^2 distances is a much less structured problem than recovering brains, so we have more reason to believe in that scenario that random bits are approximately as good as carefully chosen bits).
The main issue I am skeptical of is the statistics rather than the neuroscience. Just because the brain can be stored in 10^10 bits does not imply that measuring O(10^10) bits at random will give you what you want. But perhaps Paul has a reason to believe this beyond e.g. the intuition from the fact that random projections work for compressed sensing (which seems qualitatively different to me, since recovering L^2 distances is a much less structured problem than recovering brains, so we have more reason to believe in that scenario that random bits are approximately as good as carefully chosen bits).
I think I agree with you, but it might be misleading to talk about brain images as random bits.