In most cases, information of genealogical value obtained from government agencies, religious groups, and other organizations is provided under contractual agreements. The contracts specify what information is to be provided, how it is to be made available, and what price the web site has to pay to the provider for the records. All contracts also have a defined expiration date, typically 2 years or 3 years or 5 years after the contract is signed. In this case, Cook County obviously had a contract with FamilySearch.org.
When a contract nears expiration, the two parties usually attempt to renegotiate the contract. Sometimes renewal is automatic, but more often it is not. Maybe the information provider (in this case, Cook County) decides they want more money, or maybe they decide they no longer want to supply the data to the Web service. For instance, in the time the information has been available online, the information provider may have learned just how valuable the information really is. The information provider may decide to ask for more money or may even refuse to provide the information any more since the provider may have a NEW plan to create their own web site and offer the same information online on their new site for a fee.
Sure, that stinks for those of us who would like to have free information everywhere; but, it makes sense to most everyone else. I am sure the budget officer at Cook County thinks it makes sense.
Every contract renegotiation is different, but it is not unusual to agree to disagree. The contract ends, and the web site provider legally MUST remove the information from their web site. In this case, the web provider was FamilySearch, but the same thing also happens to all the other online sites that provide old records online.