I live in northern Indiana and have visited Chicago many times in my life, downtown, near UChicago, and both airports. I've never experienced late taxis beyond a very normal 5min sort of thing.
My late taxi experiences were always in more residential neighborhoods (Ukranian Village, Wicker Park, Lincoln Park, Roscoe Village). It isn't the land of taxi stands, but there are definitely cars in those areas, on major streets. The heart of the city is crawling with taxis at all hours, so I would not expect to see any delays there.