The government having any list of any kind of person is never good. Ever. The government already "has" a list of gun owners. 4473s are kept forever. If a gun store shuts down, the 4473s are shuttled to storage in the ATF's buildings/containers/whatever. They have, historically, used these for all kinds of questionably legal "enforcement" procedures.
However, since the 4473 list isn't entirely accurate - for example, gun transfers do not require a background check in many states, you could have destroyed the gun, lost it in a literal boating accident, etc there's plausible deniability. A government run list would certainly be designed to eliminate this - and that's a horrible, horrible thing for the 2nd. Presently the only "real list" is for NFA items. I'd imagine a government-run list would have the same privacy invading, civil rights abusing powers the NFA-owners list has.
A manufacturer on the other hand could be coerced, as "ghost gun" 80% lower manufacturers were, to narc on their customer base. However, there are legal avenues to at least slow this process down significantly. It's far better, if we have to choose, to let manufacturers have the data. They have historically been far better stewards of the bill of rights than the ATF ever has.