I have seen Why can the Discrete Fourier Transform be implemented efficiently as a quantum circuit?. This is not a duplicate.
I am familiar with the decomposition of the QFT from Nielsen&Chuang and Preskill's notes, and it requires to be able to perform n-different control phase gates. If we only allow the smallest control-phase, we can still implement all the other ones as powers but we now need to use it exponentially many times. It seems that the speedup of the quantum fourier transform is based on allowing a particularly nice (albeit reasonable) gate set.
Did I miss anything? Is there a way to decompose that circuit efficiently into a standard gate set (e.g. CNOT, H, P, and Paulis)?