I would guess that the originals for the speaker cones are likely made of laminated wood, turned to shape on a jeesly big wood lathe.
The system in the video is pretty interesting, in that it allows the operator to pressurize the area under the heated sheet plastic, to blow it out a but, then the form is raised up to it's 'working' position before the air is drawn out.
I have done a bit of free form blow molding for light aircraft windshields, and we toured through the Air Canada facility where they made a large quantity of their interior trim panels, and neither was what you could call "High Tech". Some of the Molds at Air Canada's shop were downright crude, and most were cut free from the parent stock by a not very fancy saw blade mounted on a drill press.