Warning: This is not yet a complete answer. If plausibility arguments make you uncomfortable, stop reading.
I will consider a variant where we want to multiply $(x - a_1) \cdot ... \cdot (x - a_n)$ over the complex numbers.
The problem is dual to evaluating a polynomial at n points. We know this can be done cleverly in $O(n \log n)$ time when the points happen to be $n$-th roots of unity. This takes essential advantage of the symmetries of regular polygons that underlie the Fast Fourier Transform. That transform comes in two forms, conventionally called decimation-in-time and decimation-in-frequency. In radix two they rely on a dual pair of symmetries of even-sided regular polygons: the interlocking symmetry (a regular hexagon consists of two interlocking equilateral triangles) and the fan unfolding symmetry (cut a regular hexagon in half and unfold the pieces like fans into equilateral triangles).
From this perspective, it seems highly implausible that an $O(n \log n)$ algorithm would exist for an arbitrary set of $n$ points without special symmetries. It would imply that there is nothing algorithmically exceptional about regular polygons as compared to random sets of points in the complex plane.