That version is slightly better than what exists today, but it doesn't make improper retention of information by government physically impossible. There is a way to do so and insure truly anonymous background checks (and I actually wouldn't object to universal background checks if done that way) but if they wouldn't even go for that one, they obviously wouldn't go for a real one. Since, when you offer them what they claim they want (UBC) without the thing they say they don't (back door registration) and they reject it, you know they're lying. Course, they're politicians, so that's just to be assumed.
Do you include specify that a person selling a gun may ignore any a denial that doesn't demonstrate that a person was convicted of a felony under due process of law? No reasonable informed person can believe that gun grabbers wouldn't seek to use background checks to deny the rights of people who haven't even been accused of having actually committed any crime whatsoever, given that they've overtly stated their intention to do precisely that.