Sorry with all due respect that is not true that the entire sea is moving around the vessel. "In deep water, a wave is a forward motion of energy, not water. In fact, the water does not even move forward with a wave. If we followed a single drop of water during a passing wave, we would see it move in a vertical circle, returning to a point near its original position at the wave's end. These vertical circles are more obvious at the surface. As depth increases, their effects slowly decrease until completely disappearing about half a wavelength below the surface." See http://www.onr.navy.mil/Focus/ocean/motion/waves1.htm
The wave is an energy passing along much as a waveform in a rope held at one end. The water in the wave simply moves in a circle so near the crest may be moving in the direction of the boat and faster, effecting steering.
Where the boat picks up acceleration is primarily from gravity as it moves down the waveface, which is itself moving.
I suggest that broaches generally begin at the top of the wave when the stern is not square on, and so one side is caught before the other starting a turn. Once started this is almost impossible to correct, and at the trough the bow also hits obliquely turning it further and slowing the boat sharply. At that point you need to be able to turn the boat rapidly which a drogue may tend to slow down. The drogue would also at that point be to one side requiring a greater boat movement to realign it.
While it has it uses in ocean settings and can help align the boat and prevent pitchpoling, I doubt that it is desirable in crossing a bar, or in making an inshore approach as outlined in other posts.