Everything you mention is true, but that's an artifact of Illinois being fiscally irresponsible. Isn't Gov. Rauner still waging war on unions and still hasn't presented a budget? Priorities man!
The home rule provisions in the Illinois constitution allow Chicago wide leeway in setting its own tax rates, and in keeping the revenue from such taxes. Chicago has mostly itself to blame for its looming bankruptcy, even though the long train of awful, corrupt governors hasn't helped any.