Hi Sue. Unfortunately yes, it seems to be a "dead system". There is no way to accurately diagnose this over the Internet. You have tried all the usual self-help remedies. (battery, fuses etc.). It could be a bad APIM or a handful or more other causes. If the vehicle is out of warranty, there is nothing that can be done now, other than pay to have it diagnosed and repaired. Or purchase an aftermarket Audio system that will give you Bluetooth connectivity, etc.
Just out of curiosity, you purchased a used vehicle with a non-working SYNC system. When did you first discover it did not work? Did the car have any sort of CPO or Dealer warranty? if yes, did you ever bring it back to the Dealership to have it repaired during the warranty period? We have to assume not.
Keep us updated and good luck.