I'm very old world, and one thing I love is the old proof house markings on guns, a formal and beautiful thing. I wouldn't mind the full run of proof marks on all my guns.
In any case, the bigger issue in the US isn't the fact that there aren't stamp markings and numbers for numerous parts, its the fact there is no registry of any type nationwide that is truly fully traceable. Outside of city, local, state registrations, there is no national registration, only ATF lists of who bought which gun brand new. The second someone sells or trades the gun the trace turns from official numbers on paper link to a person to a pounding the pavement and phone calling of people to try to track down the gun from owner to owner., unless it ends up at a gun shop or other FFL licensed seller.
This means that if you bought a rifle back in the 1980's the ATF MIGHT have you on record as the original purchaser. The records might outdated, lost, not submitted, burned up, beyond finding. Even if they did, and you sold the gun to a private party two years after you bought it, and then that guy sold it to another guy, and he traded it for another gun to another guy, and he sold it to someone else, the ATF would have to find your record first (which they might not) then call you, then you have to go "Umm, yeah, didn't like that gun, sold it to some asshat at the gun range 30 years ago, can't remember his name, maybe Joe Blow" then they have to find Joe Blow, then they have to ask Joe Blow what he did with it.
That's assuming some criminal or bad actor didn't destroy the serial numbers on the receiver. That's assuming they can find the record. What happens if you have dementia, what happens if Joe Blow died 20 years ago and nobody remember who he traded or sold the gun to, cold trail? What happens if any one person on an ownership chain decides to lie or obstruct in a way they can't prove he's lying or obstructing? What if you never sold the gun but told the ATF you did, how are they going to prove it? They can't.
Gun registries, or really any registries, rely on immediate and complete compliance, followed by continuing compliance. If there are breaks in the chain, there's no way to prove a gun has been legitimately lost or stolen or not. Destroyed or forgotten. Trying to track down guns to owners is legitimately impossible because owners had the right to sell their guns to anyone they didn't suspect to be a person prohibited from owning firearms. Many American gun owners trade and sell guns to buy other guns, leading to the trails becoming sloppy, even if everyone cooperates the trails are hard to track down and is time consuming. Now try tracking 400 million guns in this type of environment. Impossible!
One complaint is that this system helps protect bad actors who are straw purchasers of guns for criminals, the straw buyer can just say "Ah, my gun was stolen, I didn't sell it to Johnny Crackhead" and makes gunrunners harder to catch. On the positive, it makes it impossible fr the government to track down legitimate guns in good hands to take them away.