The ADA does quite a bit more than "only require that an employer make reasonable accommodations for a disabled person to be employed." The reasonable accommodation part is not even the most integral part of the act. The ADA broadly prevents covered employers from discriminating against persons with disabilities. This includes directly asking an applicant what illnesses they have. The act also addresses housing and transportation discrimination. The employment thing is only one part of the law.
Also, I'm sorry that the state of California's monitoring program is so restrictive as to make it so "the employer must be told the nature of your disability," though I doubt that is indeed true. However, in some other states, monitoring programs do not impose those restrictions. Here in CT, there are no rules against night shift, overtime, or specialty at all. There is no narcotics restriction unless the program director decides that it is suitable for a given participant. Many people never have a keys restriction. I was a diverter and even I was allowed to work without a narcotics restriction. For those of us in my circumstances, why would an employer need to know I'm in recovery prior to the offer of employment?
I can only think of one reason they would want to know -- and it is to discriminate.