Proposition: A measured query is not a coherent oracle

The function $f$ is classical data, but its required query operation need not be a classical channel. Replacing the coherent query by an address measurement is generally incorrect.

Proof. For $m=1$ and $f(x)=x$, $O_f$ is CNOT and sends $|+⟩|0⟩$ to a Bell state.

Even for $f=0$, the correct oracle is the identity. Measuring its address before answering dephases that address. On $|+⟩$, the identity and dephased outputs have trace distance $1/2$.