Yes, you can "self study" for the written and when you think you are ready you can take the written test without an instructor's sign off. And you are correct, if you pass the written you will have a report saying you passed the ATP written, nothing more.
You can take the checkride in a PA-34 or BE-55 and get an ATP with a multi-engine. There are many schools that will do this, 3 or 4 training flights and a checkride. And you can then say you hold a FAA ATP. In order to have an FAA ATP with an A-320 rating you have to take the checkride in an A-320.