I think bills passed with a simple majority had a long waiting period.
This is correct. The legislative session runs until May 31 each year. The IL constitution requires that bills passed between June 1 and Dec 31 require a 3/5 majority, except for those bills having an effective date after June 1 of the following year.
Does this absurd sounding rule serve any practical purpose?
Yes. The legislative session runs for only 5 months of the year. It's nominally a part time body as are many state legislatures. Anything other than that regular session is a special session to deal with emergency matters or vetoes issued by the Gov. For emergency matters the constitution drafters in 1970 felt that it should take the same supermajority vote to pass a bill as to override a veto.
However, if a bill's start was postponed until after the end of the next regular session it only needed a simple majority. The reasoning is that the bill was presumably not of an emergency nature and the next session could make changes in the law before it started using the full deliberative period.
