It could absolutely be done as NICS: rather than a phone call, plug your form 4473 info into a website. Once they do the check, it emails you the code which is

It could absolutely be done as NICS: rather than a phone call, plug your form 4473 info into a website. Once they do the check, it emails you the code which is good for N days. Your seller then just verifies the code on the ATF website and checks that your ID matches the name for the code. It would be more convenient for everyone, including the ATF employees.