"Undoubtedly those in the know can answer this better: why don't they have at least one strap to hook a foot when running a rapid? Seems like this would add more stability and control."
The reason there is no strap is that to be able to turn a board the size of an SUP, you can't just stand in the same place the whole time. To turn a board (any size) you have to focus your weight over the fins, and then when you pick up speed down the line you move your weight towards the nose (ever seen someone hanging ten?) Its a little more subtle on a shortboard but all surfers follow this concept from bottom-turn to top-turn.
Anyway, obviously on a river in rapids you could get by standing in the same spot much easier than on an ocean wave, but if you wanted to take it into play holes and actually stay on the board, it wouldn't be possible to connect more than one trip across the wave face if you stood in the same spot.
If you go to you tube and search for a video of longboarding, you'll see what I'm talking about. Turns are above the fins, and speed sections are toward the nose.