Incidentally, I don't think they need a warrant for this process (maybe they'd get one anyway, or maybe they need it for a specific trace?). You agree as an FFL to allow inspections regardless, and support traces. You even have to have a certain number of computers around, be able to print on demand, etc, etc.
You've also kind of hit the nail on the head when it comes to the regulatory motivation behind stopping private sales. The traceability of a firearm from its manufacturer to where it is now is totally provided by FFLs, and falls apart once a private sale happens. I can confirm that I can't even look up past transfers that I've facilitated with ATF using the NICS system (except for the tx number, date, and status), the only identifying records I have are the ones I keep. They (ostensibly) delete identifying information from the background check history, and they do not collect a serial number during the normal background check process.
Also incidentally, the last private transfer I facilitated (via gunbroker) involved a seller from Cali who was very, very diligent and insistent about keeping his own paperwork so he'd have answer for ATF if they showed up . Not that you're required to, afaik .
Edit: this provides some insight into how a trace starts (with a manufacturer, importer or wholesaler): https://www.atf.gov/firearms/national-tracing-center