If your house is broken into and your valuable items are stolen, you would file a report, with the model and serial numbers with the police. Also, for insurance purposes, you have to be able to properly identify what was stolen. You should have a detailed list of your valuables with make, model and serial numbers and include pictures and provide that to the insurance company. They bet on their customers not claiming everything stolen because they don't have or keep records and can't remember. This includes clothing also. When you are upset, it is hard to remember. so make a detailed list and take pictures.
But, in most cases, if you can't properly identify your stolen items by serial number, there is a good chance you will not be able to recover them.
I just know that in the case of stolen items, pawn shops, etc. if the police track items to pawn shops, they will use serial numbers. I mean, there are lots of Rolex watches stolen and pawned. Which one was yours? prove it. That is what will be said.
The OP did the right thing. Don't take any chances. There is no such thing as a free lunch. Report and be safe. Conscience isn't the issue. It is really about protecting yourself.
enjoy