When I was actively pursuing Chicago research, I sometimes searched the ProQuest version of the Chicago Tribune when I wanted contemporary insights into the workings of, say, the vital records office.
Organizing files earlier today, I came across a copy of an 1878 article titled "The County Building" which included a paragraph suggesting that the County Clerk was a bit timid about taking action to force ministers and physicians to report births, marriages, and deaths.
THE COUNTY BUILDING. Chicago Daily Tribune (1872-1963); Mar 8, 1878;
ProQuest Historical Newspapers Chicago Tribune (1849 - 1986) pg. 8
If you've been looking for a record that doesn't seem to exist, this might be why. My feeling, though, based on thousands of searches, is that most marriages and deaths were probably recorded (although sometimes marriage licenses were obtained but never returned) but that births were often overlooked.