I'll do you one better.

We should regulate barrels.

A gun rifled barrel is the only part of a gun that can't be reliably and descretely be made at home. You can make your own ammo essentially from house supplies and spent brass casings. With the upsurge of small desk lathes, CNCs and 3D printers you can also fabricate most parts of the receiver at home (which they now regulated too).
But a barrel however, you need very specialised heavily machinery to make a rifled barrel in a way that makes the gun not blow up in your face or be able to hit the broad side of a barn.
The barrel would also tell you a lot about the gun owners habits. If somebody goes through several barrels in a few years, they are probably using the firearm a lot and are therefore well trained and capable.

In my opinion serial numbers should also be stamped or laser engraved on the barrel not the receiver. People would also think twice before filing off the serial number and ruining the structural integrity of the barrel.
Technically you can slap together a home made rifle, as long as it uses a manufactured barrel it will be somewhere in the realm of precision of a fully manufactured firearm, reliability however may differ.

That's why countries like Germany legally define a "firearm" as "A device that expels a projectile through a barrel". The receiver or more precisely the mechanism of operation is herby completely irrelevant (there are exceptions for muscle powered devices).

This would also call for a revision on the idiotic suppressor regulation, which is in essence limiting the use of a device to further protect your hearing. Instead you either register a barrel with a fixed muzzle device or a barrel with a threaded end and you can put whatever you want on it. The government is happy, because it can assume you are a dead silent killer with a scary silencer, and you are happy because you don't have to wait 6 months for a glorified propane tank on the end of your gun.

Some people collect guns like Pokémon cards. Imagine if you can get 1 to 3 barrels and then 40 different receivers without having to register any of them. A lot of AR platfrom barrels are interchangeable, so you can have as many colors, setups from as many different manufacturers as you want. In the end even if somebody decided to go lulu and shoot up a place, they can't use more than 3 of them at once.

Collectors would also profit since now they can have as many historic firearms as they like as long as they all have a dummy barrel. No need to destroy the beautiful mechanics of the gun to deactivate it. And if they decide to actually fire it for demonstration, they can register a separate real barrel.

Edit: English isn't my first language so I mixed up threading and rifeling. Fixed that.