There's only like ~55 billion us banknotes in circulation (according to uscurency.gov). It wouldn't surprise me to find out that banks' counting machines scan each of them, and put the serial number and location into a database, and that database flags bad serial numbers and things like "this serial number is also claimed to be in a vault 1000 miles away" - causing the bill to be flagged, set aside, and turned over to the secret service.

The working set of data needed for this type of thing could probably be stored in a couple TB - small enough to be in a single (beefy) server's RAM.

Yeah, that was my point: these rules aren't really preventing counterfeiting, because even if you were allowed to print currency on a home printer, it wouldn't work, because it would be trivially obvious as fakes. It sounds like you are saying that making trivially obvious counterfeit bills is still possible, which seems like it even further supports the fact that these rules aren't very useful.