To me, the "paradox" seems to be:
An equal amount of water is goes up the yellow supply tube, and then down out the nozzles.
Equal amounts of water in opposite directions -- so it seems obvious that the net force is zero and it can't work.
Once that central paradox is resolved, then little details like steering and underwater operation are easier to understand.
There are 2 things that break the "equal and opposite" symmetry, each of them providing some lift:
thrust from net change in velocity
Personal watercraft such as the WaveRunner and the Jet Ski and this JetLev take water in through the intake and then out through the jet nozzle.
The water going out the nozzle is much faster than the water coming in the intake,
so the drag from the intake is more than compensated for by the thrust from the jet nozzle.
The net difference in speed of the water
(from still lake water to relatively fast-moving exhaust water moving in one direction)
leads to a net acceleration of the stuff pushing against the water (in the opposite direction).
This effect would still happen if the pumping was done solely on the flying unit with a rigid pipe into the water.
Decoupled thrust on boat vs thrust on backpack
While the pump on the boat is off, the water on the lake "comes up to" the painted water line on the boat -- i.e., that's the point where the downward force on the boat (gravitational force on its own weight) balances the upward buoyancy force.
The pump on the boat accelerates approximately stationary lake water upwards.
The reaction force pushes the boat downwards, driving the painted water line underwater.
Eventually the boat settles to a (lower in the water) position where the downward force on the boat (gravitational force on its own weight, plus the reaction force from pushing water up the tube) balances the upward buoyancy force.
Meanwhile, the piping at the top of the takes the high-speed water coming up the tube and turns it around.
The change in direction of velocity leads to a net acceleration of that piping (and the backpack and pilot to whom it is firmly attached).
In a sense, some of the force of gravity on the person is transferred to the boat, pushing the boat deeper into the water as if the person were standing on that boat, even though the person is not directly touching the boat.
It's vaguely like shooting a water cannon straight up, and balancing a ball centered over that cannon.
This system also has similarities to the "Lofstrom loop".
It helps me visualize the forces involved if I picture the water as chunks of ice that move around.