This is interesting!
I have a paddle wheel on my boat. I know it works, because I can turn it on and spin the wheel, and something shows up on the screen. How accurate it is has yet to be determined, but at least I know it works.
And I would tend to agree with you that the paddlewheel or something "instant" is going to be the best way to tell that "pulling that rope dropped my speed"... You would need to check the speed just before and just after the change you make (and only change one thing at a time!) to see the impact, but over the course of the 30-60 seconds you're testing something new, your environment shouldn't impact the speed that much. Also, when I was looking through the manual for my old paddlewheel, I believe it said you could change the sampling rate, meaning you can adjust how quickly the display updates with new information from the paddlewheel...so you can tune it to give you a more instantaneous readout, which you wouldn't want during "regular" sailing.
The GPS would definitely be the better way of determining whether you are being consistent with your sailing...good sail trim for 20 minutes, then you messed something up, you were wandering and not maintaining a steady course, etc.
And don't forget, the admiral can ALWAYS tell when you've messed something up!
By the way, yes, pitot tubes can be used for water (for any fluid really), but they and the readout have to be calibrated because the specific gravity of the fluid makes a difference in the accuracy (and they have little holes that can get plugged up). There are also other methods of measuring speed, by using devices designed for flow measurement (check out Dwyer Instruments' website) such as thermal dispersion, etc. However, they are designed to measure in feet-per-second or something like that, which you'd have to convert. I still think the paddlewheel is the easiest choice for instant gratification...