The next step in the process, actually used to make bearing balls, and claimed and probably applicable if you really want spherical shot - which "flies straighter," is to roll the shot between rotating steel plates. A few modern shot makers reported use of this process, and had very proprietary (and sometimes colorful) names to distinguish their shot from that merely "whipped in shape by the wind":
Picture two circular flat plates laying flat with one above the other, the top plate having a hole in the center but with the centers of the two slightly displaced. When an "almost round" bunch of objects is poured into the hole, with the plates rotating, the plates are of course pushed apart by the diameter of the larger objects. As the plates rotate, they "squash the big chunks" until they contact all of the chunks, and the chunks roll along between the plates. Because of the circular path, and the displacement between the axes of the plates, the "chunks" are "spun" as they roll, so that they come out sperical rather than as cylinders. The amount of displacement between the axis of the upper plate and that of the lower plate determines how rapidly the "chunks" migrate to the outer rim of the plates and fall out - by then being "perfectly round" and all (at least statistically) of precisely identical size.
Some of the modern shotmakers have touted that the rolling process also "microforges and hardens" the shot. It's somewhat questionable whether forge-hardening lead is of real benefit for lead shot, since it "self anneals" rather rapidly; but it is a very real and important factor in the quality of ball bearings - and the ad department has to claim every benefit they can imagine, since someone might believe them.