So based on recommendations, I stopped at the Cozy Dog Drive In on the way to St. Louis.  
Cozy Dog Drive In
2935 South Sixth Street
Springfield, IL 62703
http://www.cozydogdrivein.com
We'd already eaten earlier (my traveling companion hadn't had breakfast, so we had an early lunch at a subway -- her choice), so the two of us each bought only a Cozy Dog.  The first ones we got were cold in the middle on the first bite, so they fried up another for each of us.  So what's a hot one like?  Nothing overly special, but worth the stop.
The cozy dog was a typical, non-kosher-style frank: soft, bland and greasy in texture.  The breading is fresh-dipped and fried to order, has a great texture, and is somewhat sweet and a hint of spice.  The nearby counter has relish, mustard, ketchup (horrors!), hot sauce and "Route 66 Herb Vinegar."  I chose a bit of vinegar, which added some tang.
The place itself is an History Route 66 Attraction, with lots of kitch: signs, "artifacts", and souvenirs such as keychains and plush Cozy Dog "beanies" and a Route 66 library (touch books only after asking permission).  Overall, prices are very low, with the most expensive item #3.39 for a double or chili cheeseburger (there is a $12.99 item for a dozen Cozy Dogs)
Worth a stop?  Sure, it's only a mile or so off the highway.  A must-stop?  Probably not.
That evening, in St. Louis, we were lucky to find an unreserved table downtown, as the Cards had just finished pulling off an extra-innings sixth-game series-tying win in the playoffs (what, somebody's still playing baseball when there's none in Chicago?).  Caleco's (visible from my hotel window) was an OK bar & grill & Italian sorta place, with a sauce bolognese that would be shameful to show up in Chicago.  Pretty good wings, though.