Making that information public would probably be illegal and a huge violation of privacy. Making the database of serial numbers of stolen weapons available might be a good idea, as long as the person that the weapon belonged to agreed to it.
It is the serial numbers that could be made public. AFAIK, the current system only allows for a search by serial number, NCIC number, or another identifying number.
It is an old legacy system. I do not think an owner’s name is entered into the system, just a case number at the entering agency.