As it stands the firearm information other than cursory does not pass to NICS, and they already have more information on the potential buyer than is on the 4473. The minor benefit of not alerting the central database when a check is run (which is already destroyed in the current system) is far outweighed by security and information integrity issues of having thousands of copies of a database full of personal information.
In order to have an accurate list you need a lot of identifying information. It's not as simple as, well I searched John Smith, and it said someone named John Smith is A-OK. It has to be the right John Smith, which in some cases would take a lot of cross-referenced information to verify, which paints a pretty detailed picture of someone, including SSNs. Having all of that information on thousands of computers across the country is a security nightmare.
Even if you created a system where people where people were approved by a negative result you'd still have all the information on there for the offenders whose right to privacy and information security is not diminished. Additionally, a negative result system would still have to be smart enough to require additional verification for conflicting, incomplete and/or inconclusive information erasing in some cases the minor benefit of not alerting a central system.
Whether or not I approve of NICS is irrelevant, what you're suggesting is not an improvement on the current system, its a step towards more issues. I'm sure no identity thief would ever want a database containing millions of people's identifying information...