You can just look at Redfin to see that this isn't true. You can get a 4bdr in the Loop for ~$800k. I've spent most of my life here, including the last 19 years, and I just bought a new house here; I think you're going to have a hard time talking me down from my own lived experience.
The standard message board tactic for wriggling out of this position is to define "the nice neighborhoods" in Chicago ludicrously narrowly, like the only nice places to live here are Lincoln Park and... well that's it. But even Lincoln Park is doable at $800k.
The median condo is not particularly great - and it goes for ~$660k in most of the nicer neighborhoods.
The median house is well over $800k in the nicer neighborhoods.
And, again, Chicago is one of the most affordable cities in the US.