In a decent sized quantum Fourier transform, almost all of the phase gates are rotating by extremely tiny angles. For example, in a thousand qubit Fourier transform, 99% of the phase gates are rotating by an angle that is less than 0.000000000000000000000000001 degrees. That amount of rotation is irrelevant. You can skip 99.9% of the phase gates and the output will be de-facto indistinguishable from the ideal output. There's really a surprising amount of flexibility in the phasing.