Example: The whole gate set does not force dyadic probabilities

On two qubits initially $|00⟩$, apply $H$ to the first, then controlled-$H$ from the first to the second, then $H$ to the first. The amplitude of $|00⟩$ is

and its probability is

Thus exact dyadic sampling is a designed subroutine, not a universal property of every circuit over the frozen alphabet.