That crossed my mind but I figured water is coming at this area from many directions
Well, I think that's a debatable proposition, and certainly one we should be able to confirm from the local geology, but it would depend on the area in question. Just to be sure are we still talking about the Grand Canyon? Because the only source of water I'm aware of in that area is the Colorado River, which exhibits exactly the qualities I described.
If one rivulet could get started across a flat surface others also would
They would quickly input into the first rivulet, which would become the main channel of outgoing water, which is exactly the sort of drainage pattern we see in watersheds to this day.
I've wondered if it would be possible to set up an experiment by spreading out flat wet clay in a field and letting it harden a bit and then run water across it or sprinkle water from overhead to see what happens.
That's a great idea; that's an experiment that I performed in 8th grade Earth Science so its quite easy to do. I suggest you do it in a box so you can correct for the lay of the land, and experiment with various configurations of land and lay. It should be quite illuminating. If you had a digital camera you could both easily record the experiment and share it with the rest of us.
I'm sure some scientists have already conducted these kinds of experiments. We should probably ask them.
Absolutely, but I really like the idea of performing it ourselves. Also I would reccommend you view satellite imagery of various watershed areas; its easily found on a google search.