I am interested in unit cost for mass production. It needs to be significantly cheaper than an old style top-loading washing machine to be affordable. The design of old style washing machine is mature and priced at around $100 for 8kg model. I suspect it can be stripped down further, remove water pump, remove program controlled inlet valve et al. to reduce the cost to below $50. Granted, washing machine like that needs electricity, but solar panel may be cheap enough.
One more thing, the water is not always easy to get in poor places. It is often much easier to carry laundry to a well, creak, or river than transport water to home. The path to the water sources may be a narrow trail often going up and down hills, so even with wheels on the machine, it is impractical to drag the machine to the water.
Extremely simple washing machines already exist, and I suspect on the order of 30-40 dollars. They are top-loading. No pumps. Turn one dial to let water in through the inlet. Turn another dial to let water out through the outlet valve. All manual, no pumps. Then flip switch to start spinning with electric motor, flip it back to stop spinning (no timers).
What you do is fill it with water. Add soap. Then put in first load of clothes and run it for 15 minutes. Then take out the clothes and put them in a tub. Repeat with second load of clothes in same soapy water. Once, all loads are done, then put in fresh water. Run all loads through it to get the soap out. You are done.
(Relatively) richer people might have another machine that acts as a spinner. Otherwise, you just hang up the wet clothes outside.
Of the stuff I've repaired recently it's been mechanical switches that have caused problems, where a microcontroller and bloody great MOSFET would have kept on forever.
There are much lover cost washing machines with electricity. This one for example: https://www.youtube.com/watch?v=2eo4CIHpp28 . It's just a motor, you just need a big bucket or a tub to fill it with water and detergent. Let it turn the clithes in water for half an hour and rinse. It's also handy during rinsing.
Growing up in a developing country in 90s we used to use this type of machine for bigger loads because our normal front loader machine was only 6kg capacity. This + a bathtub was the way to go for washing the blankets, bedsheets etc. It costs now sth like 20-30 usd.
One more thing, the water is not always easy to get in poor places. It is often much easier to carry laundry to a well, creak, or river than transport water to home. The path to the water sources may be a narrow trail often going up and down hills, so even with wheels on the machine, it is impractical to drag the machine to the water.