Example: An energy readout is not automatically a time readout

Take one clock qubit and one system qubit, with

Then $K|\Psi ⟩=0$. Enter in the clock energy basis produces system states $|0⟩$ and $|1⟩$, each with probability $1/2$. However, $U_S(t)|0⟩⟨0|U_S(t{)}^{†}=|0⟩⟨0|$ for every $t$; it cannot turn the first conditional state into the second.

The problem is not failure of the extraction identity. It is failure of the proposed basis to satisfy clock covariance for two distinct readings. Changing to the clock basis $|{\tau }_0⟩=(|0⟩+|1⟩)/\sqrt{2}$, $|{\tau }_1⟩=(|0⟩-|1⟩)/\sqrt{2}$, with $t_1=\pi /\omega$, produces the conditional system states $|+⟩$ and $|-⟩$, which do satisfy the asserted evolution.