Well, the thing about Stand up Paddle Boarding is that eventually (pretty much a lot) the board will tip over, getting everything wet.
Some people like to SUP on flat water (lakes, marinas, lagoons, etc.), but most SUPers like to catch waves. They get into the waves much earlier than traditional hand-paddling, making so that the surfer not only gets the wave he wants (it can get competitive out there), but can get up on a plane riding and angling before the wave gets too steep.
So, using the trolling motor would probably create too much drag once he caught the wave.
Now, there's another option: Remote Control gliders have props that actually fold flat when the plane is gliding, but when the motor kicks in they become rigid (from spinning so fast) and create thrust. The rc pilot makes his plane climb to the desired altitude and then shuts off the motor, saving precious battery power for the wings. It enables the rc pilot to fly for an hour or so (more if he can find thermals), whereas the typical rc airplane will fly for 5-15 minutes.
But what about wetness. The system has to be completely submerged and inverted several times in an hour or two of surfing.
And how about this?
It's a stick that you use to push a skateboard, enabling the SUP surfer to keep in shape on land and enjoy a different way of propulsion on land other than using one foot to kick.
What if someone attached a small hub motor to the end, filled the shaft with some lipos, and had a way to push a skateboard. It would be like an e-skateboard, except that the skateboard would be unaffected, enabling the rider to use the board conventionally as well as electrically. I wonder if that would work.
So many dumb ideas that one of them might just work.