So I was doing a bit of reading.
Apparently the obstacle to generating energy from the forces driving capillary action is breaking the surface tension at the top of a capillary tube. It is just impossible to get the water to fall to drive a generator, getting it out of the top of the tube requires more energy than you would generate.
Or is it?
Here's the deal. The horizontal tube rests on the edge of the vertical tube, held down against it by the attached weight when there is no water in it.
So, the horizontal tube is touching, disrupting, the meniscus that will form at the top of the vertical tube. Water will begin to flow down the small horizontal tube by capillary action. This is especially certain if the horizontal tube is much smaller so that its diameter is less than the height of the meniscus.
At certain point, the weight of water in the horizontal tube causes it to rotate, breaking the surface tension at the meniscus, and allowing water to pour out of the lower end of the horizontal tube.
Then the tube rotates back to horizontal.
Rinse and repeat.
And in case you think it might be complicated by the horizontal tube touching the water, this works even simpler if you reduce the size of the horizontal tube such that it never could touch the reservoir.
Best Answer
As you've indicated in your title, the correct question is "why doesn't this work". The system, as described would continue to produce energy indefinitely without any being added to it (perpetual motion, violation of conservation of energy...) . So you can be sure there's a problem.
I believe that part of the trouble here is in the assumption that the horizontal tube will allow the water to leave after it tilts down. If the capillary forces involved are sufficient to draw water up, there's no reason to think that they would not be sufficient to retain the water when the tube is rotated.
If you were to build such a device, it would probably work as described until the tube rotated down. The water would stay inside and the system would just stop.