The spirit of the law you cited includes one very pertinent fact:

Further, such a list, record, or registry has the potential to fall into the wrong hands and become a shopping list for thieves.
I know way too much about information security, and I know specifically that the government, especially at the local level, is pretty bad at it. Also, the physical security of where the files are kept could be questionable. A hacker of middling talent could bypass the commodity firewall (username: admin password: 12345), tunnel into the file server, snatch the Excel sheet with all of us dastardly NFA item owners - boom, shopping list for some high-value firearms. Bill Smith over in Miami Shores has a machine gun? Hmm...