Sales from an FFL are already exempt. This would only apply to face-to-face sales, which by it's very nature implies used guns are being sold.
So no, the serial number wouldn't help catch people skirting this law.
Agree it wouldn’t be useful anytime soon, but todays new guns are next years used guns and Democrats plan for the long term. To quote the late and unlamented former Democrat mayor of Atlanta, “our goal is a society where only the police and military are armed.”