Sure, that’s true. In that case it’s kinetic energy you’re extracting, and due to the incompressibility of water there has to be a change in the cross-sectional area. But that’s not directly related to the original potential energy of the water; it’s just the means of extraction.
Still, consider the river as a whole. Keeping things simple, there’s an inlet and an outlet. Since typical rivers don’t have their outlet going at hundreds of meters per second, it must be that almost all of the energy was lost in between. Most of that could be described as some form of friction.
To extract energy, we have to reduce that friction somehow. In some cases, this is done by simply skipping the river part. You divert the flow from upstream, send it through smooth pipes to a turbine, and then outlet downstream.
In other cases, you build a dam. The cross-sectional area becomes huge and so the velocity and thus friction go way down. You can then extract energy either by potential or kinetic means (i.e., either using the gravitational force to turn the wheel or letting the water accelerate first and then driving a turbine).
But what of a wheel just placed in the flow? It still has to reduce the friction somehow. The effect might be minor but it’s still present. And it can’t have more than a fairly local effect. It must be that it’s slowing the flow in its vicinity, and for that to happen there must be a change in cross-section.
Well, the gravitational field stores energy. It pretty much has to if you’re to maintain conservation of energy, though one could just view it as a bookkeeping measure (especially since it has the strange property of being negative). And it can get complicated and ill-defined in general relativity. Still, it’s there, and gravitational waves wouldn’t exist without it.
Of course, the gravitational field doesn’t generate energy; it just stores and converts it. But then, the same is true of everything.
There is a lot of physics going on during the course of a river’s journey. On a gentle slope, a river is in laminar flow with much of the flow happening near the middle. In laminar flow, the loss of energy is minimal.
In the rapids or riffles or waterfall phases, not all the energy is converted to heat. Some of it is used in eroding soil or rocks, picking up rocks and silt and moving them.
That’s the opposite of the question the OP was asking. The OP was asking, if the power plant is gaining energy, then something else must be losing energy, and what is that. The energy that goes into building the power plant is not the energy that ends up transmitted down the wires.
Gravity gets its energy in the same way that a spring gets energy when you stretch it - you’re adding energy into the system by pulling the ends of the spring apart ‘charging’ the potential energy until the spring is released then it becomes kinetic energy.
Same thing with two bodies that are attracted to one another - you ‘charge’ the system by separating the bodies (i.e. lifting something up on Earth)