"Interesting question I hadn't considered, as I'm generally of the mindset that the decision to run off under bare poles would come simultaneously with that to deploy a drogue…"
In my experience, it's rarely been a case of deciding to run off under bare poles, but rather the situation building to that point, so by the time I might want to set a drogue, it might be too late to do so safely without coming up and slowing the boat. However, if conditions are so extreme one is running under bare poles and wanting to stream a drogue, it might not be the most prudent thing, to come around and lay her broadside to the wind and seas, even for a few minutes.
"As I also mentioned, my boat happens to lie ahull quite nicely, at least in conditions that have not become too extreme… So, even lying ahull for the brief period of time might be sufficient to get the drogue in the water. Once everything is hooked up, it shouldn't take more than about 30 seconds, after all…"
When does anything on a boat take anywhere near the estimated time to actually accomplish the task; 30 seconds is going to be at least 5 minutes.
Quite often when I take way off our boat (like tacking in extremely light air) and have a fish line in the water, it becomes imperative that we attend to that line, lest it become tangled in the wheel or rudder. I can see exactly the same thing happening if one is trying to stream a drogue in heavy weather without sufficient headway.
"It you're still flying a bit of headsail, you should be able to round up sufficiently to achieve at least some sort of fore-reaching attitude, that will at least reduce your speed."
Would you consider doing this in 30 to 40 foot breaking seas? As you have said, each boat is different, but these discussions should consider the least seaworthy of cruising boats, not those exceptional vessels capable of taking the really bad stuff with a grin. I've laid ahull in something around 70knots; just pulled up the board and secured ourselves below to get some absolutely needed sleep. No problem at all.
However, there are numerous boats sailing the seas today that just should not consider laying ahull, no matter how tired the crew may be. Those are boats that most likely won't have any fittings that could take the load of a drogue, either. What can they do to get through these conditions safely? I know many of us think they shouldn't be there, but we also know they are going out there every day, and they are the ones badly in need of a good plan. I've been fortunate enough not to have encountered extreme conditions in a boat I didn't trust, so I can't tell you what I'd do. But when I was doing hurricane season deliveries from the islands to south Fla, I'd keep the Bahamian islands close at hand, so we could make a hasty retreat and get ashore, before a storm hit us.