Where does this idea come from? If this is the case, the document being discussed in the OP would not even exist, because there would be no need to document situations causing unintentional discharge.
It seems to be acquired through social osmosis, i.e. hearing people talk about it and then repeating their assertions as if it were a fact - usually with subtle changes colored by the speakers own world view - until some version of "truth" permeates society. I guess you could consider it like a game of telephone at a societal level. The Citizens United case is another great example where what people think it held is very different from what it actually held. It's also frustrating given the fact that we live in a time where the barrier to verify these claims is incredibly low or even non-existent because of the internet.
People who just dislike guns (or dislike the people who like guns) then repeat it forever, even after proven wrong.
I think your examples make a good point though. There are absolutely times and places where someone's right to gun ownership can and should be suspended or even outright denied - the same as we do with freedom of speech, voting, or any other right. But it's already illegal in every state for recently released felon to own a gun - only 12 states have some sort of automatic renewal of gun rights and most of them are 10 years after release for non-violent offenses only.
It depends on your definition of "mentally unstable" but there are several mental health questions on the form you have to fill out for the ATF if you buy a firearm. Paradoxically in the 50s-70s there was a huge push to deinstitutionalize mental health treatment, close the vast majority of mental asylums, and make it much harder to involuntarily commit someone, but that's also the primary mechanism by which you prevent someone from acquiring a firearm on mental health grounds.
It's been a while since I've done a deep dive on background check requirements as they tend to vary by type of firearm but you cannot buy any firearm from a dealer without a background check. Honestly I think this is the one gun control advocates could make the most ground on, and the one that people can most reasonably disagree on. Quick AI math tells me that 32 states allow the purchase of a rifle from a private individual without a background check, which covers about 48% of the US population.
[0] https://x.com/GIFFORDS_org/status/1953570096443510896 claims nearly 1,000 people die from gun violence in Colorado in a year when the average is less than 300 and it's never been more than the low 400's - in fact numbers have only drastically increased as Colorado has passed more and more restrictive gun laws
Anti-gun propaganda.
Anti-gunners first started practicing lawfare against companies involved in the gun industry by suing them for crimes committed using products the companies were involved with, even though this is a ridiculous idea never applied to any other type of product. They do it because, even when the lawsuits are thrown out eventually, they are still very expensive to defend and most companies in the industry don't have particularly deep pockets.
Gun-rights advocates got laws passed in a number of jurisdictions clarifying that gun manufacturers are only liable for genuine flaws in their products. This has largely squelched these types of nuisance lawsuits.
Anti-gun activists don't like that, so they frequently spread false information that such laws prevent manufacturers from being liable for genuine flaws in their products. Despite such false claims, no law ever passed or proposed actually does this, and no law in this area has ever been repealed due to actually blocking liability for genuine flaws.