Dear RichD:
Navier Stokes cannot be solved in general. So you are correct, it cannot be solved analytically without many assumptions.
As am I. It would look no worse than some models... just depends on where you place infinity (how deep and wide and long the river is).
"Intuitively" is commonly wrong. It is in this case also, unless the "edges" can balloon out to "provide the illusion" that the water can escape. But the balloon will eventually fill, and momentum will be lost around the edges still.
I'm not sure how the little "winglettes" on the wing tips of aricraft are working out. That is the type of job they are supposed to perform also...
Keeping the paddle fully vertical, and making it travel just a little bit faster than the boat, will make it lose less water around the edges, minimize vortexing, and keep the momentum transfer as much along the line-of-ship-motino as possible. But it will cost in mechanical losses.
I don't find an easy answer. I did find this historical reference, where two identical ships, one paddle, one propeller had a race... URL:
David A. Smith