Think of how a jet boat works. A jet of water coming out the back of the boat pushes on water to move the boat forward. A fluid pushing on a fluid.
I don't think that's how jet boats work. Isn't the exit nozzle above the water line? The boat pushes water backward out of the nozzle; this pushes the boat forward. It doesn't matter what happens later to the expelled water.
If the boat is going downstream it will go faster than if it would be in still water because the jet of water is pushing against water that is also pushing against it. I know there's also less forward friction on the boat when it travels downstream so if that’s an issue for you imagine a water jet powered hovercraft instead of a boat. Going upstream if the water was flowing at the same speed as the jet of water the hovercraft wouldn't travel upstream at all.
If the jet of expelled water flows backward at the same speed as the water in the river, the boat isn't accelerating the water backward at all. So the boat will not be pushed forward, I agree. But the reason is not that the water has nothing to push backward against after it leaves the nozzle; the reason is that the water doesn't push forward on the boat while it's inside the pump.
