"The problem with this assertion is that it doesn't seem to follow from any numbers that we have. Now, it's very hard to find research that is non-biased one way or around, but playing with raw stats published by various countries and states in US has led me to conclude that gun control or lack thereof simply does not correlate with crime and violence in any meaningful sense."
You'd be hard pressed to draw a link in large part because it takes time for the effect to follow through. In the UK it took the likes of Operation Trident post-ban to work hard to remove guns from criminals (guns that could trivially be replaced if still legal, even if only in other parts of the country). It took a decade but it's worked, gun homicide now is in the tens each year, many of which are murder suicides/suicides with legally owned shotguns or hunting rifles. Gun crime is still hovering around the 10,000 mark which sounds bad, but becomes laughable when you realise that nearly all of those incidents are nothing more than kids dicking around with air rifles and BB guns which now fall under firearm crime classification for reporting - this is something worth bearing in mind if you're evaluating UK stats as it will create disparity in seriousness of incidents with nations that do not include what are often just seen as kids toys by many in the stats.
Our country is at the least violent it's been since crime stats began, possibly ever. The stats that are highest are things like sexual assaults, but these have been grossly distorted in recent years by literally thousands of reports of historic abuse mostly in the 60s, 70s and 80s by everything from Catholic priests to TV presenters. The rate of actual incidents now is also down, even with increased support for reporting such crimes. If it's correlation you want, the UK most definitely has it over the longer term, but finding causation is likely going to be an impossible task.
I don't even think there necessarily is a strong causative link, much of it is smart policing policy, but I think it's reduced the incidences and seriousness of some crime. I know I don't need a gun to protect myself, I know I'm not going to get shot and the worst thing I'd ever have to do is run from a knife. In other words whilst gun control can't really entirely be given credit for reducing the UK's crime problems it is a part of it - removal of access to guns made it possible for police to start going into estates which were simply too dangerous before. It's a two stage process - the ban on guns, or certain guns removes the legal supply of guns, which removes the ability for the legal supply to feed the illegal supply, that creates the conditions to start actively targeting and reducing the illegal supply and it's that that has the impact on reduction of crime but that reduction will take years to achieve to a reasonable degree, especially with gun ownership to the extent of that in the US and so it'd take a strong long term political leadership to stand there until the results filter through in the face of people saying it's not working after only having given it a year or two.
That has happened in the UK however thanks largely to political consensus on the issue, but also partly that we're an island and we are now reaping the benefits. Of gun crime involving illegally owned weapons it's time and time again the same physical gun popping up, the fact gangs are having to share only one or two weapons to commit crimes limits the amount of crimes they can commit quite drastically and when the police do seize that weapon or two it can leave entire gangs without a firearm at all. Sure at this point it may descend into stabbing to a degree, but at least stabbings don't tend to effect innocent bystanders like stray bullets do.
I agree it's a complex issue, and I agree the benefits of stronger gun control are often oversold as a miracle cure - I don't think that, but I do think they're at least an important part of the package.