While I appreciate your efforts to apply the three criteria of judging causation to the question of gun control, I think there are other variables that must be taken into consideration in evaluating the first criteria. Even if the intention may be present regarding the desire to kill another human being, regardless of whether there is a gun present or not, it is difficult to argue that the presence of guns make the commission of a crime far easier than many other weapons. Stabbing or harming someone in a fist fight can be deadly, but it is much easier for someone to coolly and impersonally use a gun to commit murder from a distance.
Perhaps a better way of phrasing the cliche from the point-of-view of an advocate for gun control is: 'guns don't kill people; people kill people -- but guns make it easier for people to kill MORE people very quickly.' Particularly with the creation of semiautomatic firepower and other forms of guns which can be quickly fired off, it is difficult to imagine the modern phenomenon of 'school shootings' or shootings in other crowded places without the use of a powerful gun.
I think the issue of correlation, in other words, should be phrased as: "is there a correlation between increases in violence and violent deaths in the absence of gun control and greater availability of firearms?" That is the question which much be answered. Given that the U.S. has a very high rate of homicides compared to other industrialized nations with similar socioeconomic profiles, I think there is a great deal of evidence of an established correlation between gun availability and violence. However, critics will contend that there are other aspects of U.S. society that are not present elsewhere and could be causative factors in generating a higher violent crime rate, such as socioeconomic equality and the disaffection of young people.
If Madonna said it
It is true that not every person who has a gun uses it in a violent manner and many people who use guns do so in a responsible manner. Gun control advocates, however, usually suggest that the availability of quick-loading guns such as semiautomatic weapons, make it much easier for would-be criminals to commit crimes. I agree with the idea that a longitudinal analysis between the availability of guns and rates of violence would be illuminating in terms of evaluating U.S. policy regarding firearms, but it would also be useful to compare U.S. policy to other nations that do not have similarly liberal firearm policies, to see how their rates compared with the United States. It would be also interesting to see if other violent crimes NOT using a gun resulted in fewer deaths, indicating that even though there will always be violent people in the world, the availability of a gun makes it much easier to engage in violence.
Gun control advocates do not argue that people who commit crimes with guns have no responsibility. However, the aim is to make violent crimes less apt to spiral out of control, and to hem in the ability of violent people to use guns. That is why appropriate background checks are also an important foundation of gun control: to ensure that the people who get their hands on guns will use them for legitimate purposes.
But on some level, I think it will always be difficult to either prove or disprove the link between guns and violence, given that so many other sociological factors can affect violence that pro-gun activists will use as an argument against gun control legislation. The U.S. is a complex society facing many social challenges, and violence has always been a part of its history. Availability of guns would seem to escalate this tendency, but there will always be other factors that pro-gun activists point to as causation, including violent video games as was the case after the Connecticut school shooting.