Those behind most legislation want a national registry and they want to reduce ownership.
Some states have universal background checks. I am in one of them. There is no private sales or transfers. All transfers must go through an FFL, except to child or grandchild and those need what is essentially a registration form sent in.
The result is it is essentially a crime to have guns you are not on record as purchasing, and almost not have guns you are on record as purchasing.
There was some extra like those owned before various dates, long guns, etc
They have passed legislation registering all long guns transferred by 2014 too. And all people entering the state will need to register and pay for each gun they bring.
You have FFL transfer fees. In addition to the cost of the gun, you get to pay a third party money just to transfer your property.
Because going through an FFL is a requirement you don't get to opt out of such fees.
Perhaps not a big deal on more expensive purchases, but on $100-$200 guns having to pay another $35, plus gas, plus purchaser wait 10 day waiting period to even test out the used property...
And they don't stop, its never enough, they want to expand it, and do. They make programs to use such registries, now they have the automated computer system that red flags you if you get prohibited and results in a team of armed cops showing up well funded.
At the same time various legislators at the state level propose and pass legislation to add to the list of misdemeanors and other offenses to make someone prohibited.
It is a nightmare, they are emboldened and only get worse once they have a better idea of where every gun is. That starts with a requirement for paperwork on every single transfer.
As for the extra information on a 4473:
It is retained by the store and if a gun turns up it is traced from manufacturer, to distributor and retail location, and then they use the 4473 to determine who it was sold to from there.
So if they get a serial number they can determine who is on the 4473 for that firearm.
It is still an obvious risk, and often ends up filed away with the ATF for future use. Something mandated if the business goes out of business within 20 years, and many do. Also records may be taken during various investigations leading up to that, and some stores and chains create thier own electronic databases of customers data, which in investigations large portions or the entire database of customers may be copied and retained in mass by ATF etc even more readily than the paper 4473.
However because of private transfers even that information if gathered is still not a reliable registry in states that allow them. Because in such states without a required background check those on file as purchasing a gun 10 years prior may not have that gun, and don't have to answer to anyone for it as it was perfectly legal to sell it without involving government.
While in places requiring government permission, aka background check, every transfer should have a record and tracking down all legal guns or punishing those that cannot account for them is easier.
(They also add requirements to report lost or stolen guns in 2-3 days in places to add an extra punishment for those that try to say they had a 'boating accident' as the common expression goes. If they come for the guns you could legally only claim to have lost them within a couple days, or have committed a crime, and you are unlikely to have lost them right as they decided to show interest or want them and such a claim would be unbelieved and may even amount to probable cause to some.)