[retroencabulato]

I'm confused; I don't see how the randomness is really important here. I modified the code such that the expected value of each coefficient is a function of the term's degree:

    randomPoly[n_, x_] := x^Range[0, n].Table[RandomReal[{1, i}], {i, n + 1}]

With this bias (or any of the various of biases I've tried), the result is exactly the same.

I guess my observation is 'symmetry' of the coefficients has nothing to do with the 'symmetry' of the roots.

[Sharlin]

I think coefficients that are merely linear (or even polynomial!) functions of the degree are all "almost equal to 1" in this case. As user conistonwater mentions in another comment, "I also think it's important to mention that the roots ax^300-b have magnitude (b/a)^(1/300), which is much closer to 1 than b/a, which is why the magnitudes of a and b don't matter very much."

[pfortuny]

Sorry I did not reply sooner. My answer what rather vague. Symmetry meant something related to the disk unit and zero and infinity but in a very very imprecise way...