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It's interesting that you chose a random magnitude-phase pair rather than a random real-imaginary pair. The points are therefore not uniformly random over the complex plane right?
Since the roots and coefficients of a polynomial are Fourier transforms of each other, another way to rephrase the question that I find interesting is to ask why the Fourier transform of a random real vector is another random vector most of whose components have unit magnitude. I don't have an answer as to why but maybe this will help someone find another intuitive explanation.
