First of all, it's not "my" theory. It's the accepted theory of how rivers cut their canyons. I'm not sure I can give you a clear reference for it though, because it's essentially common sense.
Second, the Colorado River is hardly a "narrow stream". Perhaps in relation to the Mississippi, but it's still 175-700 ft wide and 9-130 ft deep within the canyon. That's a lot of river. Also keep in mind that it's restricted by Glen Canyon Dam, so it isn't receiving its full flow anymore.
So now to explain how river downcutting works. As the river cuts down through the rock, it behaves differently when it reaches different layers. A hard, resistant layer (like the Kaibab Limestone or Coconino Sandstone near the rim of the canyon) will be cut in more-or-less sheer cliffs, much like the base of the canyon is now. However, whenever a softer layer (Hermit Shale or Supai Formation, for example) is reached, the river is able to erode much more quickly, and also broaden. It undercuts the more resistant layers, which collapse into the river and are eventually swept downstream. In the course of this, it loses some energy (this can also be caused by raising sea level and/or subsiding the drainage area of the river, lowering its gradient), so it may have trouble cutting into the next resistant layer it encounters. The river then will change into a meandering course and continue to widen its floodplain, undercutting more and more (which explains the big shelves in the canyon). Then, base level (sea level) drops or the drainage area is uplifted and the gradient of the river increases, which increases its energy, allowing it to downcut again.
Hope that explained things.