This does not at all ring true. I mean, yes, you could not take all of Chicagoland tomorrow and cram it into the city limits. But that's in part because a fraction --- a significant fraction, and probably the majority --- of suburban residents chose to live in the suburbs because they did not want to live in the city.
I live in Oak Park, just across Austin, because when we moved here we were concerned about the schools in the neighborhood we wanted (Lincoln Square). We had no trouble finding houses to look at, though! (I sort of regret the decision we made, though I like where we live now.)
In fact, even in the most desirable neighborhoods in the city, it's straightforward to find housing in Chicago. People do not move to the suburbs because there's no room in Chicago. There are cities that have that problem. And Chicago has a lot of problems! But it does not have that problem.
I live in Oak Park, just across Austin, because when we moved here we were concerned about the schools in the neighborhood we wanted (Lincoln Square). We had no trouble finding houses to look at, though! (I sort of regret the decision we made, though I like where we live now.)
In fact, even in the most desirable neighborhoods in the city, it's straightforward to find housing in Chicago. People do not move to the suburbs because there's no room in Chicago. There are cities that have that problem. And Chicago has a lot of problems! But it does not have that problem.