Drops are better (imo) for road riding as they offer you more positions, you can ride on the hoods, on the bars or on the drops so gives you more chance to move around for comfort. You say you don't see many people on the drop, that is because for 'general' riding they are probably the least comfortable of the three positions, also on the hoods is more aero if you keep your arms flat. The drops though offer much more control and I would not want to be anywhere else on tight bends especially on descents. Also in a bunch sprint you have no business being in any other position from a safety point of view.
From the point of view of the video, I have only watched a little, but he will I imagine be on a specialist lightweight hill climbing bike where every gram that is of no use is got rid of. The drops are quite a bit of weight saving and not much use whilst climbing, and as I doubt he had any fast tight downhill bends to ride, no use for them.