I highly doubt any U.S. immigration official will get upset that a mistake was made in the city of birth of a U.S. citizen petitioner. It was not wilful misrepresentation and it is not material misrepresentation.
Also seconding what SalishSea said. A birth certificate taking months to arrive is not typical in the U.S.
When I ordered extra copies of marriage & birth certificates for my state, they all arrived within a week. Also, applying in person for a birth certificate can take as little as half an hour to get the certificate. So, for hopeful immigrants reading through this thread, please do not conclude months of waiting for a simple birth certificate is what you are by default doomed to experience.
I looked up Illinois's Dept of Vital Statistics website and it says requests by mail for certificates are taking 12 weeks for processing. No specific information on how fast VitalChek takes to process.
https://dph.illinois.gov/topics-services/birth-death-other-records/birth-records/obtain-birth-certificate.html