You may not call it a registry but there is most likely a definite paper trail that will lead to the last person who jumped through all legal hoops with the exception of states that still allow FTF transactions.
Not quite. It can trace only to the first person who bought the gun from the FFL. Even if that person transfers the gun to someone else via an FFL, the government would not be able to find that record unless the person in question remembers what FFL was used and provides that information to them. Without that, the government wouldn't know which FFL (or which FFL's records) to go to in order to find the next owner even if the next transaction took place via an FFL.
There is a paper trail. It leads to the FIRST person who purchases from an FFL (maybe not even that if it was long enough ago, before the 20yr provision was changed to forever and the FFL in question stayed in business long enough to be able to legally destroy the records without turning them over). At that point, unless the state in question has registration, the trail ends unless that owner is obligated (or chooses) to keep and provide records on the next transfer of possession.
Which means that there is not even a partial registry. There's literally no information at all about who currently
owns what, and only a partial record (that currently can't be digitally searched legally) of who
purchased what at one point. Except in states who set up their own registries.
For there to be even a start at a registry, here's what would need to happen:
1. All FFLs would be required to forward 4473 information to the feds immediately.
2. The restriction preventing the ATF from compiling 4473 records into a digitally searchable database would have to be repealed.
3. The law would have to change to require ALL firearm transfers to go through FFLs.
That would create an effective registry of all guns sold in the future as long as only legal transfers took place under the new system. It still wouldn't backfill the information for all the previous sales although it would get them closer to being able to trace a gun to a current owner.
To get a complete registry, in addition to those 3 changes, the law would have to require current owners to register their firearms. With all four changes, assuming compliance, there would be a reasonably good registry of all legally transferred firearms.