The water analogy seems to imply that power = current. Why is this incorrect?

Power to a water-wheel depends both on the current (amount of water delivered) and the head (vertical drop of water as it turns the wheel). So, the water analogy does have TWO variables that multiply together to make power: current, measuring (for instance) the water flow at Niagara, and vertical drop (like the height of Niagara Falls).

Current is NOT the same as power, in a river, because long stretches of moving water in a channel don't dissipate energy as much as a waterfall does. Siting a hydroelectric power plant at Niagara Falls makes sense. In the analogy to electricity, a wire can deliver current at little voltage drop (and has tiny power dissipation) but a resistor which has that same current will be warmed (it has a substantial terminal-to-terminal voltage drop).

Here is a simple way to keep this stuff straight.

Power is always the product of an effort variable and a flow variable. In hydraulic systems, the effort variable is pressure and the flow variable is the flow rate.

For flow in open channels, the effort variable is typically very small (but not zero) and the flow variable is very large. BTW power exchange which occurs at low effort and large flow represents the low-impedance regime.

In your water example power cannot be equal to current because they have different units (power is an energy per unit time, while current would be something like a number of particles passing through a surface per unit time).

...the more water that flows per unit of time, the more energy gets delivered to the wheel per unit of time

What you have noticed here through your analogy is that power is proportional to current (as an example, the more force you apply to an object, the larger its acceleration, but this does not mean that force and acceleration are equal). In a circuit element this proportionality is the voltage, since it tells you how much energy is associated with a "unit of current". You would need a similar way to convert your water current to the power generated by that current (although this might be a simplistic model of how power is generated using a water wheel).

You also have to keep in mind that it is an analogy, and all analogies have imperfections. With the water analogy, power is generated by water actually pushing on a wheel. In circuits, $P=IV$ is much more general and applies to any charges undergoing a potential difference.