Have you requested a copy from Cook County? I requested a death record for my great-grandmother that I couldn't find in the index, but they found it and sent me a copy. Turns out the name was badly mangled in the index.
Also, are you sure he was actually born in Chicago? Is it possible he was born elsewhere and moved to Chicago at a young age?
Do you have a copy of his social security application? That would have a birth location, and should match the birth record he used to apply for a SS number.