You would think that the FAA would at least offer to contact the owner. They don't have to share the contact info. Sure, it's only $5 to register a drone, but for what purpose? What service do they provide to the owner of that drone?
The
purpose of registration is to facilitate finding the owner who may be responsible and liable for damages in the event that their property (the drone), and actions (flying the drone/putting others at risk) disfigures a toddler or worse when the drone crashes. The $5 registration fee pays for establishing and maintaining the registration database, and using it for enforcement actions that benefit our society. No ”service” to the registrant was ever implied or intended.
If you fly for recreation, you fly under so called ”community-base rules” like the AMA’s. I don't know of any other community-based rules so the AMA’s may be the defacto standard. I believe that the AMA rules require an ID label on the aircraft that includes your AMA membership number, and/or name, address, and phone number (I assume if not a member). Again, this is not really to facilitate recovery of a lost aircraft.
Commercial flying regs also have aircraft ID requirements.
Getting the FAA involved may not be doing the owner of the subject lost drone any favors. Losing control of an aircraft isn't trivial and may lead to an investigation and some sort of legal action, assuming the FAA has the resources and interest to follow up. Someday they will. I assume the owner will at least have to file a report with the FAA. If the report says ”it just flew away and I don't know why” isn't going to help the owner’s situation or the drone industry.
If you fly, it is possible if not likely that you will lose a drone at some point. If you don't know this, maybe finding a different activity would be wise. In any case, in addition to your registration number, putting your contact info on the drone to facilitate recovery without having to involve the FAA is probably a good idea.
When you fly, you put others at risk, and you are responsible for the damages/injuries you cause. Therefore, it's also a good idea to fly with drone liability insurance. AMA membership includes $70k of insurance - assuming you are flying within their rules. I'm sure their insurance does not cover flight activities outside their rules. Otherwise, you can use the Skywatch.ai or Verifly Drone apps to buy on-demand flight insurance. Again, these providers will not cover activities outside the applicable rules of your flight.