Sure, let's give an overworked, underfunded, and frequently despised Federal agency even MORE to do! That's a good idea, right??
I realize that it sounds like a good idea, but there are at least two things you aren't taking into account. The first is the GIGO principle (garbage in, garbage out) meaning that the performance of the system is limited by the quality of the data. And the second thing is the law of unintended consequences.
Both of these will act to make the proposed system difficult to work accurately, and ultimately turn it into something much different from what was originally envisioned.
Think it will be easy to fix errors in the data once its in such a system? I think that's overly optimistic. The CORRECT data might not even be in the police files, computer, or paper!
I recall hearing about a case from a few years ago, where a fellow had to defend himself against the charge of having a stolen gun. IIRC, Ultimately he was cleared, because his gun was bought new, from a dealer, and he had the paper train to prove it. But it still took time, and multiple meetings with police, etc. The gun in question was a S&W .38, ser# xxxxxxx.
Which was all the information the police had, that a S&W, cal .38, ser#xxxxxx had been reported stolen. The innocent owner's gun was the same make, caliber, and serial number as the one reported stolen. It was, however a different model gun with a different barrel length.
The police had information, but not enough information, in their records. There are guns out there with the same serial numbers. Some are even the same caliber. It has happened, and WILL happen again, especially in a computerized system fed bits of data from all over the country, with NO WAY to check the validity of said data, short of a time consuming, and expensive investigation of every piece of data entered!
This isn't just a "minor flaw" that is "easily fixed". One of the worst cases of unintended consequences has already been mentioned. SWAT team coming to "visit" you because the system sad you had a stolen gun.
When some adrenaline amped kid just a few years past his Academy graduation, wearing body armor and a machinegun, guns you down in your front yard, because he thought you made a "furtive movement", who benefits??? Your heirs?? MAYBE, but maybe not. After all, the officer was just doing his job, etc...
No, I have to put the idea of yet another Fed gun database, even one "open to the public" as a poor idea. Best of intentions, certainly, but all to easy for it to become something with bad results.