Correlations DO in fact establish the point and sociologists since the foundations of modern social sciences from the time of Durkheim and Weber onward have established this point. And Habermas's contemporary work on both communicative action and Normative standards and the foundations of Law reiterate it over and over. Crime is almost always an over-determined act socially speaking and laws can very seldom be tied directly to the commitment of crimes. You cannot, for example "prove" that laws against murder slow the murder rate. Most legal arguments can only be meaningful against the backdrop of certain kinds of lifeworld (lebenswelt) assumptions or certain kinds of ideological constructs. As a result - in most cases correlation is not only the best kind of proof the social policy-maker has it is often the ONLY mechanism they have.
Your distinction concerning correlation and causal connection is simply wrong according to the mechanisms of most decent
social science and philosophy. It is the discourse of science and mathematics. Even though some social scientists have attempted (I believe with no success) to establish Coefficients of Correlation, these are not only too complicated but are not meaningful to normative and social discourse. This is clearly one of the things that Habermas established in the Theory of Communicative Action and one of the things that he has long fought against concerning the colonization of social discourse by technocratic discourse. Incidentally Feyerabend also did brilliant work in this area.
Connections between certain kinds of social behaviour and certain laws and social policies is in fact correlative not causal, period. Furthermore, the neo-conservative efforts to colonize all social policy making with absolute causal 'proofs'and quantifications is part of a nearly global and entirely disingenuous effort to create the appearance of necessity and technocracy, all the while being driven by fairly firm ideological efforts.
Of course, an issue like gun violence is over-determined and cannot be reduced to any single policy or social phenomenon. But if you need to get technical then this much is clear. No law, NONE, can be said to be causal since only individual actions by independent actors could appropriately be referred to as causal. Thus the great sociologists like Habermas have established normative discourse in terms of a complex web of determinates. Since one cannot "prove" that loose gun laws in the US have lead to the highest rates of gun crimes in the world, the sociologist is dependent upon another kind of discursive exercise. This is not "illogical" or faulty logical as your original comment implied. Rather it is a kind of normative discourse that depends on correlations and certain kinds of communicative actions. And since you, and no one else, can actually prove a causal relationship between any law and certain behaviors (a fact even many right-wing theorists have been at pains to establish) it is entirely disingenuous to demand such proof in normative discourse. Rather, it is a different kind of correlative and communicative mechanism that is demanded. I can no more 'prove' by technocratic standards that a gun registry will reduce gun crimes than you can 'prove' by the same standards that generous welfare causes the working class to be shiftless and lazy. And since no such mechanism of 'proof' exists you can choose to believe that there is no link between the loose gun control mechanisms in the US and the highest rate of gun violence in the world. There is in fact no causal link, properly so-called, only a correlative one.
We know, for example that in countries like the UK and Holland gradually established stronger mechanisms of gun-control, including registries, and gun-violence (particularly domestic violence involving guns) gradually declined more or less in line with the establishment of such controls. This is not causal, it is correlative. But hard logic and technocratic reason can still let you deny such links. But technocratic rationality doesn't actually belong in the realm of normative discourse and in my opinion technocratic rationality is largely and ideological illusion anyway, a position made clear by a long history of skeptics from Montaigne to Feyerbend. You failed, when you talked of the “empirical” question, to understand that I was in fact arguing that you cannot, in almost all cases, make such empirical connections within a region of normative discourse. In other words I am using an entirely different paradigm of discourse rather than the 'empirical' one. Furthermore, I am arguing that I explicitly reject the paradigm you are using because it is a technocratic colonization of normative discourse, which is not only useless but entirely disingenuous. And I am building this argument on the back of Weber's work on rationalization in Economy and Society, Horkheimer and Adorno in Dialectic of Enlightenment, and Habermas' work in Action, Theory of Communicative Action, Theory and Practice, and Between Facts and Norms. (See also Luca Corchia's work on Explicative Models of Complexity)