It's immensely intrusive. Moreover, the NICS system isn't adequately funded, or enforced, as it stands; requiring the entire population to be checked by the system, and their records to be kept up to date, would be very expensive indeed.
The only people that could possibly have a problem with this would be criminals who do not wish to be found, and those whose real purpose was the database they wished to abuse in the first place.
"The innocent have nothing to fear" is a line that's been used by governments for a very long time in order to justify restriction of rights, invasion of privacy, and denial of due process. Typically, the innocent find out that they did have something to fear, but by then the damage is done. For example, consider the U.S. government's 'no-fly' list for air travel: it's easy for a completely innocent person to be listed, and just about impossible to get off it.