Remark: The quantifier matters

The proposition concerns the full clock–matter input space. It does not forbid preparing a fresh clock in a known basis state from a classical code. Indeed, on the block-diagonal class

a channel can reconstruct the displayed density operator by preparing $|{\tau }_k⟩$ conditional on $k$. This is recovery of a classical sector, not recovery of arbitrary clock coherence and not return of the consumed names.