Proposition: Disorder-averaging counterexample

An averaged channel can be constant while every realization retains distinguishable logical information.

Proof. For a qubit, let

Their equal mixture is the replacement channel $\mathrm{Tr}(\rho )I/2$. For each realization, the outputs of $|0⟩$ and $|1⟩$ are orthogonal. By the triangle inequality, its distance from any replacement channel is at least $1/2$. Thus averaging has erased information in the description, not proved its absence in each realization.