Also, there is already a strong sentiment in all non-Chicago parts of Illinois that Chicago should be it's own state, because it dominates all statewide politics, gets the lion's share of resources, and has a tendency to mess up calculations for needed rural aid for the rest of the state.
Calling the State "Chicago" would certainly add fuel to that fire.
I'm originally from Illinois (Non chicago) and I would agree with that sentiment. In fact, where I'm from, everyone called it the People's Republic of Chicago and the rest of the state as Kentucky proper . (As a joke of course) On a more serious note though it is frustrating from a political perspective when Chicago dominates stae politics so much. For example, the city of Chicago sued my town, because there was an obscure provision on our books that allowed a certain tract of land to be Section 8 housing. So the state jacked up our taxes to build cheap apartments and then literally bused people from Caprini Green (A project) to our sleepy little town. This was all done in the name of lowering Chicago's crime, not by fixing it, but simply moving it outside the city. I'm sure it had something to do with their failed olympic bid. (Another issue our town had to deal with)