Ref the post just above, I’ll point out to our British OP that there is a Springfield in Illinois, between Chicago & St. Louis. There is also a Springfield in Missouri, past St. Louis on the same highway going southwestward, now numbered 44. They are comparably sized. Mr. **Cell Guy **is talking about Springfield, Illinois.
Back to the OP & my earlier comment about traveling in August …
If you’re not attached to the idea of seeing either Route 66 or the Southwest for its own sake, you might have a more enjoyable and interesting time just proceeding west from Chicago and ending up Seattle rather than Los Angeles. And then driving south along the Pacific coast to end your trip in San Francisco as you’d planned.
Still plenty of small towns & amazing emptiness & scenery. But not quite so hot & desolate. OTOH, a lot of the Northwest is boreal forest, which you may have seen in Germany or Scandanavia, or even parts of Scotland. Conversely, the southern plains & desert southwest are utterly different from anywhere close to home for you. Depends on how exotic your taste runs and how happy you are in 100F /40C or hotter weather.