One travelog video I was watching claimed McCormick Place sells overnight RV parking in their marshalling yard (where truck and rail cargo is loaded and unloaded)... you prepay and get a reservation ticket from a website?
McCormick is a bit south from the Mag Mile, etc but it is A LOT closer than a KOA or something like that - only a few blocks to public transit (bus, l-train, etc) and then the city is yours.
It's just a secure parking space, not a FHU rv resort. Bring your own power and water, let the City do the cooking.
Regarding your other stops, there's a bus stop at every other block as I recall up and down both sides of Michigan Ave, as well as LaSalle I think? Sorry, it's been years since I was there last. I'm sure you can find a stop closer than Big John to the Driehaus.
Google shows a bus route stopping right near Erie, and the red line stopping 3 blocks north or 3 south of Erie.