There is no good reason to record this info. The filling out the form either pass or does not pass the background check. If the person passes what does it matter what firearm is being purchased.
I have never seen an anwser to the question, how many crimes are solved by tracing a serial number on a firearm?
Almost no crimes have been solved by tracing the serial number on a gun or by tracing a fired cartridge case. Many states (including here in FL) have no paperwork requirement for person-to-person gun transfers.
The FFL paperwork has always been part of a leftist dream to confiscate firearms.
If you have some basic machining capability, you can build a firearm on a milled out 80% lower that has no serial number. You can build a Glock clone, AR15 or AR10 clone quite easily. For the AR's you can build a rifle that will shoot sub 1-inch groups at 100 yards, though I am not quite sure how I came to know that.
“how many crimes are solved by tracing a serial number on a firearm?”
I can think of one. Sort of. 40 years ago I had a pistol stolen and reported it to the local police. I gave them the serial number and got a call 18 months later. No charges filed against the person who had it because, they said, he could have bought it from a private seller who had no idea it was stolen.