So, my 2 cents....
A few years ago, during renovations to Soldier Field, the Bears played at University of Illinois Stadium, which is down state, but holds a fair amount of people. Why can't they demo Soldier Field and build a new one in it's place while the Bears play 8 games a year (maybe 9) at Illlinois Stadium - again? For probably no more than 2 seasons.
The key is to start construction on the new stadium as soon as weather breaks - to pour foundations and do subgrade work (utilities). Once the foundation is poured and walls start going up, the weather becomes less of an issue to complete construction and ostensibly, they could move along fairly rapidly.
It's not a perfect arrangement, but it will get the Bears the stadium they want, and they stay in Illinois. And, they and their fans have done the Illinois trip before, and they have demonstrated it's not a signicant burder.
Have they completely ruled out this option?