Benevolent Government
The "mandatory training" proposal presumes benevolent government.
I can't say I'm old enough to remember FDR, but I was around for LBJ, and voting age when he got the boot.
Since then I've seen a parade of presidents, senators, congressmen, governors, agency heads, and bureaucrats of all stripes who all have one thing in common: do whatever it takes to secure their position.
Individually, these people are (generally) decent, thoughtful, persuadable people. Together, and in the context of their positions, they behave in a predictably I'm-the-boss-and-you-will-comply manner, always ready to assert authority whether they have it or not. They are public "servants" who spend their time enforcing their will on the public whom they ostensibly "serve." They do not suffer from such confining concepts as "the customer is always right."
In theory, having a (benign) government department of some kind do the (benevolent) record keeping is a good idea. I know that, if I ran a large company and had to keep track of customers and memberships, I'd have a department for that. Except that, in a company context, the department keeping the records, and the departments whom they service, would have no authority over the persons on whom the records were kept.
In government, there's always someone who has some kind of authority and someone who wants more authority than he already has. There's always somebody who looks to centralize and consolidate. I'm sure there are appropriate biological metaphors, but the point is, government is not neutral.
Government, as a seemingly natural inclination, shows a repeated tendency to evolve toward some form of socialism (or other totalitarian construct) under some banner or other, and sees individual independence as a threat to its authority and control.
As much as we would hope that a new department in government would -- or even could -- remain neutral, we know it won't happen. They simply can't permit a "low impact" solution (like voluntary record keeping) to remain that way. "OMG! How will you enforce this?! It's full of freaking loopholes! *** is wrong with you?! Don't you realize you need tighter regulation? And so it goes.
No one disputes that training is a good idea.
For people who get hunting licenses, hunter safety courses are, in fact, already a requirement in many jurisdictions.
The problem is, the thinking process that goes from "it's a good idea" to the (logical?) conclusion that "therefore it must be mandatory" is insidious and a device of creeping incrementalism.
Good idea --> mandatory is an unfortunately plausible construct that lends itself to stricter and stricter oversight and control of the individual, until there is (mandatory) surveillance in every room of the house to enforce "good ideas" like brushing your teeth, how much toilet paper you use, how high a flame you use under the frying pan, monitoring of contraband substances in the kitchen that might lead to cooking with trans fats or (shudder) butter , and a review of possibly unsafe sexual habits in the bedroom.
Yes, I understand that gun safety isn't entirely about the safety of the operator, and that there are bystanders to consider, but any measure taken to "guarantee" gun safety on the part of government is not going to follow the originally imagined vector.
If you want safety in a given domain, a better approach is a cultural one.
Our children are taught routinely in schools to be careful when crossing the street, and a whole spectrum other safety and personal care teachings.
Schools routinely teach car safety, drug safety, alcohol safety, sports safety, safe sex, and so on. And sex isn't even enumerated in the Constitution.
You want safe shooters? Bring back rifle teams. Add firearms safety to the "Health & Science" curriculum. Add gunsmithing to the auto shop & wood shop family of classes. Little Johnny gets graded on his ability to learn and apply gun safety. Gun clubs in high school. Gun safety week. I mean, really, isn't it appropriate to integrate something actually enumerated in the Constitution into our educational processes?
By the time a kid graduates from high school, the military won't have to worry that they're recruiting gun morons. Guns will be so commonplace culturally that anyone seen doing something stupid with a gun is immediately censured by his peers. "Hey, moron, rule #2. Your momma raise you in a convent?"
Demystify guns. Put them back into the mainstream of the cultural thinking.
Rather than have government track who's been trained and certified, make training and certification the default. Hey, if you got a high school diploma, you know gun handling.
Is that more effort than creating an agency for oversight? Why, yes, it is.
Still, in my estimation, it's very much worth it.