One of the challenges with using analogies like the water analogy is making sure you use the correct objects to draw your analogy from. Many answers here argue that hydraulic power depends on volume and pressure. This is true if your water wheel looks something like this:
This is a drawing of what a modern hydroelectric turbine looks like. They are designed to be able to efficiently draw power from large volumes of water with large pressure drops, such as the pressure drop from the bottom of a lake out to atmospheric pressure. In these cases, the water analogy works rather well, as expected.
However, when I think of a "water wheel," I have a different image. I imagine something much older:
These operate differently, and lead to the conclusion you drew, which is that water wheels generate power from current alone. The reason you end up drawing this conclusion is because this sort of water wheel wastes any power from pressure or velocity. The only potential it is effective at generating power from is the gravitational energy of the water from a high altitude to a low altitude. If you were to spray high pressure water at one of these water wheels, most of the energy would be wasted as the water splashes off of the blades. Some energy would indeed be transferred to the wheel, but it would be tremendously wasteful.
The water wheel is most efficient at handling cases where the vast majority of the energy of the water is stored up as gravitational potential energy -- energy from being up high. And it's best at converting water which is exactly at the height of the wheel. If you drop water down from high above the wheel it will turn, but most of the energy will be wasted in splashing and sloshing.
Thus we would treat this water wheel as a "constant voltage" device, in our electrical analogy. The setup around the wheel has the effect of guaranteeing that most of the potential of the water is its fixed height as it enters the wheel. Any energy above this is wasted as heat. And, indeed, if you look at the math, when your voltage is constant, your power is indeed proportional to the current. It's the special case where this is true.
We do indeed have devices that operate this way, but you have to enter the world of semiconductors to do so. Diodes are small semiconductor junctions used to make current flow only one way. Try to flow against it, and they're like a check valve that stops the water flow.
Well, almost like a check valve. They operate like a check valve up to a point, called the "breakdown voltage." If you put a voltage higher than this across the diode in the wrong direction, it starts to let current through. It will dissipate any energy that comes from running current across this voltage drop as heat.
So what the old fashioned water wheel is most like is a motor with a reverse-biased diode on it. Any potential of the water beyond the gravitational potential energy the wheel can handle is lost to splashing and sloshing. Any potential from higher voltages applied to the diode and motor circuit is lost to heat as the current flows through the diode. The remaining gravitational potential of the water multiplied by the volume of water sent through the wheel tells you how much mechanical power is generated by the water wheel. The voltage across that motor (after the diode limits it) multiplied by the current going through the motor is how much mechanical power the motor generates. The analogy holds, you just need a more complex circuit to model the 6000 year old device!
Incidentally, we do actually design circuits like this. In modern circuitry, we often have "zener diodes" which have a carefully tuned breakdown voltage to be a "voltage reference," and we have voltage regulators which are designed to resist the flow of electricity just enough to ensure a specific voltage across the remaining circuit.