I imagine even our rivers get their water from rain and mountain snow melt. The Animas starts essentially at Silverton Colorado, maybe 40 miles north of Durango. The mighty San Juan's headwaters are Wolf Creek Pass, also in Colorado.
The South Fork of the Rio Grande also rises from Wolf Creek Pass, not many feet from the San Juan. On a rainy day, you can see the distinct beginnings of both. When you translate Rio Grande, you do get the idea that those who named it weren't used to much in the way of rivers.
I don't know why the Lower Rio Grande would be larger than the upper. We also have the La Plata here, and it completely disappears for stretches as irrigation water is taken out, and reappears when the ditches return to the rivers. Maybe a similar effect takes place with the Rio Grande.