I think people might be misinterpreting the question.

The easy fact that I think Gil has in mind is that if you randomly choose a quantum noise operation with certain properties, then for most choices of that noise operation, although the total error rate will be very low, the errors will be highly correlated.  The orthodox interpretation is that this is an artificial way to choose a quantum noise model.

One analogous classical question is as follows:  If you randomly choose a stochastic map on the probability space of bit strings, with similar properties, what will it look like?

Gil proposes a randomly chosen permutation subject to the condition that at most t bits are flipped.  I am not sure that that is really a good analogy to the quantum noise model that he proposes.

The Poisson process that flips bits is a specific model of random noise on bit strings.  It is not a randomly chosen model of noise.