Poor people don't need LEDs? LEDs are the cheapest light source in the long run. The electricity/batteries to run it might be expensive or impossible to get in some places.
Check out this gravity-run light source which is "essentially free" and being used in many places.
Even if it did, the physics don’t really work. A high end bright LED light draws a few watts. Let’s say the LED uses 1 watt, and the gears are perfectly efficient. One watt-hour is 2655 foot-pounds. Assume the weight is 265 pounds with a 10:1 pulley to lift it. You’d have to pull the rope attached to the pulley 100 feet to charge the light to run for one hour, and the device would need to be 10 foot tall. Figure in mechanical losses, and a more realisitic LED, and you end up multiplying the weight, and pully ratio by ten. Now, the weight is similar to a small car, and you pull the rope 0.2 miles for a one hour charge. Make it 5 foot tall, and the weight and rope double again.