Tldr? Why would anyone do this? To what end?
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You can do this with a long enough hose and height difference by just using river water
Why not set something up by the river instead and just route the power?
That requires a much bigger river, that carries enough water to drive a big ass wheel, which limits your options for location a lot. If you take a look at the video i linked in the neighboring comment you can see what is possible with just a little water stream if you have the right setup.
To utilize power that is otherwise wasted? The water can still be used in the garden etc. Also if there is a power cut the water often keeps running, so you could use this to charge a small battery or a phone.
If the power goes out you'd probably be better off using solar or a even gas. Maybe if you're using water for something else anyways it makes sense but how much power can really be generated by turning a house on for a min? How much water is wasted?
Given most of water comes from public facilities it is structurally just unsanctioned and ineffective transmission of power from shared pump to private generator. Would be a simple energy theft if you didnt pay for water. A waste of water and energy anyway.
That actually kind of makes sense. It just seems like there's a reason we pay for these things.