The major reason that reality diverges from image is that Interstate Highways have been built over much of the original right of way of Route 66 - especially in the West. What that means for cycle tourists is that they have to ride either on an interstate shoulder with 20,000 vehicles zooming by or on a service road that offers little respite from the roar of the interstate. Idyllic it is often not.
Furthermore, Route 66 had many routings over the years - i.e. there is not ONE Route 66. This is true of many of the named highways of old such as the Lincoln Highway or the Dixie Highway. Given this, I would argue that the preferred choice for the cyclist in the Southwest would be to approximate Route 66 - to experience the natural beauty, native cultures, and some of the cultural artifacts of Route 66 in a way as close as possible to that of early cross-country travellers - rather than to adhere to any fixed route.
For example, from northern New Mexico to the Grand Canyon, I believe it is far more rewarding to take a route such as Taos, Abiquiu, Cuba, Gallup, Window Rock, Second Mesa, Tuba City, Grand Canyon rather than follow service roads and interstate shoulders from Albuquerque to Flagstaff. The former allows one to experience Taos Pueblo, the art of Georgia O'Keeffe, ancient pueblos, the kitsch of Gallup Route 66, Navajo life, the traditions of the Hopi, and finally, a rim ride along the Grand Canyon.
Just sayin', ya know?