Why do the unitary gates on the measurement qubits have $2^n$? Why do we need to apply the unitary gates for any power at all? What would happen if we applied the controlled-$U$ only once, for example, for each control qubit, prior to the inverse QFT?
For example, from Wikipedia here.
Specifically the circuit diagram illustrated in Wikipedia.
Another resource that shows this is the Qiskit documentation.