The Illinois Division of Insurance (DOI) site lists that the question of if one can keep a vehicle after it is declared a total loss by the insurance company as a common query. Their answer to this is that it depends upon how old your vehicle is.
The DOI states that to minimize "chop shop" crime, Illinois law lets you keep a totaled vehicle only if it is nine years old or older, or if the vehicle has incurred only hail damage that does not affect the operational safety of the vehicle. In that case, your insurance company may, but is not required to let you keep your vehicle. If you have a newer vehicle, you must give the vehicle and clear title to your insurance company before they can settle your claim.
So according to information given by the DOI on the insurance fact sheets, your insurance company is correct that you cannot have your vehicle back due to IL state laws. If your car was 9 years old or older the insurance company may have allowed you to keep the car but likely you would have had to pay them salvage cost to keep the car.
You would not have received blue book or actual cash value plus get to keep the car. You would have received your settlement minus the amount they believed they could have received on the car for salvage since they would no longer be receiving this amount. Salvage cost is typically what the insurer would have received from the junk yard or an auction at which they would have sold your vehicle in order to try and recoup some of the money they paid out on your claim.