It says that it's no kids allowed, but...
He named teenagers as welcome.
They have a high chair.
They have no kids menu.
None of this says to me "no kids allowed." Maybe there are special rules for dining in Chicago (and I don't say that facetiously) but I'm not sure you can just ban children from places that aren't gyms, adult theaters, private clubs, or bars. They have civil rights and stuff. But making it as uncomfortable as possible to bring strollers in probably will make it more hassle than its worth for parents to bring them.