Because in my mind I take a "random" polynomial and only consider the largest two roots. Then I consider doing two reflections: across the bottom/top of the curve, and across the x axis. If those top two roots aren't degenerate the combination of the four curves using the reflections have two curves with a largest real root, and two curves with a largest imaginary root (I think). If it is degenerate then the largest root is real.
So I would then conclude that (1) there are more real largest roots than imaginary and (2) the "advantage" is vanishingly small since there are infinitely more curves with a largest non-degenerate root than degenerate root.
As one can tell I barely know the proper nomenclature. I assume I'm mainly missing some consideration about the uniformity of random coefficients not resulting in a uniform distribution in space? Or possibly my reflections can violate one of the conditions (real coefficients?)? Or I'm just plain wrong.