I have not checked timetables closely but here is an example of a similar trip that could be done by Amtrak that goes to the best of the places on the Route 66 tour, and more, and provides more time to actually see things. Amtrak has a 15-day rail pass which allows 15 travel days and 8 travel segments during a 180 day period, $389 for an adult. I would not compare Amtrak to the best of UK and European trains on which I have traveled, but Amtrak trains are comfortable, have dining cars with quite good food, snack bars, beer, wine, mixed drinks, dome cars for viewing, bathrooms. And the regular coach seats are quite large and actually comfortable to sleep. There are also bedrooms available on the Southwest Chief for extra charge. The best thing about being on a train is that you're not trapped in your seat and can get up, move around, make friends, chat.
So here's my "stab" at an Amtrak "Route 66" trip.
Fly into Chicago - stay 2-3 days; museums, shopping, lakefront
From Chicago take Texas Eagle train to St Louis - 1-2 days in St Louis; Gateway Arch, more museums
Take Missouri River Runner train to Kansas City to connect to
Southwest Chief train to Albuquerque to connect to
New Mexico Rail Runner (not part of Amtrak) to Santa Fe 1-2 days
NM Rail Runner back to ABQ
Southwest Chief to Williams AZ (from ABQ to LA the rail route parallels the old Route 66)
Grand Canyon RR (not part of Amtrak) to GCNP; 11 AM; overnight stay at GCNP; GCRR back to Williams
Southwest Chief to Los Angeles, 1-2 days
Coast Starlight to San Francisco, 2-3 days
There are special rail travel planners in the US, and one may even already have a similar trip including lodging reservations. There may even be already rail journeys similar.