My family history used to be cut and dry...My Mom was adopted to a good home. Taken in by the dentist of her biological mother, an 18 yr old immigrant from Ireland who was impregnated by her employer——a much older prominent Jewish man from Hyde Park in Chicago. As I’ve become older, I feel this story isn’t true. My Mother was born at Cook County Hospital in Chicago, IL. She says her birth records are sealed in a way I’ve been unable to imagine how and why; the legalities; how do you get through life without a birth certificate? is this possible? Is it impossible to ever discover the truth?
In many states, yes, original birth certificates (OBCs) are sealed when a child is adopted. An amended birth certificate is then issued with the adoptive parents' names and the child's new name. So, no, adoptees do not go through life without a birth certificate.
You're somewhat in luck; Illinois, according to the American Adoption Congress partially opened their adoption records. Adoptees born 31 Dec 1945 and earlier can request a copy of their OBC. For adoptees born 1 Jan 1946 and later, they can file for a non-certified OBC (it can't be used for identification), but birth parents, if living, can file a veto and have their information redacted until their death. More information from AAC here.
Of course, there's always the possibility that your mother's birth mother did not name the father on the OBC, or used a fake name. She may have even used a fake name for herself. And falsified the non-identifying information she provided to the adoption agency. So getting a copy of your mother's OBC may not actually provide any decent information.
If you really want to find out who your mother's biological parents are, I recommend going the route already suggested: Take a DNA test. Even better, if your mom is still living, have her take it. There are a couple books by Blaine Bettinger that are great guides for selecting a DNA test (or tests) and interpreting the results. You can also ask further questions here, or there are a number of DNA/genetic genealogy groups on Facebook (including some that specialize in adoption/uncertain parentage cases).