Hello TURIN,
You are of course correct in that the casing isn't actually permitted to expand to its full un-cooled size, and then caused to shrink by blowing air over it. But the effect of the cooling is to keep the casing at a smaller diameter than it would have been without the cooling air.
I'm less sure about your emphasis on delaying expansion of the casing to permit the turbine disc to catch up. If it was simply a matter of delaying expansion then the cooling air would be used only for very short periods and not in any steady state condition such as cruise flight.
I'm not familiar with the specific details of the CFM56, but if ACC is used in steady state cruise (as some of the posts in this thread appear to suggest), then it cannot simply be a matter of allowing the turbine disc to catch up with the casing. The first few minutes of cruise should do that.