I'm in the PNW so we don't have a lot of overlap in native species, however I can recommend selfheal. Selfheal is a great ground cover that's native to much of the US (including Cook Co). If you let it grow it gets to be about a foot high, but you can also mow it and it'll stay more grass height. It produces little purple flowers and is evergreen in my area (but probably not yours). I haven't had any luck getting it to self propagate or germinate from seeds I've spread in the yard, but the plants I've germinated in pots (super easy) and planted in the ground have established and are still doing great years later. In other words, you can grow in amongst grass ala clover, or let it grow as a standalone plant.

But I guess that leads me to my main question, which is do you want to forego having a "lawn" entirely, or do you still want an area that can support foot traffic, but populated with lawn-substitute plants (eg native grasses and grass-companions like clover and selfheal)?