The mere fact that uncontrolled influences are present makes what the op wrote fail as a disproof, it does not actually matter what they are. If you cannot control all factors such that a single example blows the argument, disproof by counterexample does not work. This makes it mostly useful in math, and less so in the real world.
As the saying goes, correlation is not causation. A lot of people have real difficulty with this, especially when the correlation supports the argument they are trying to make, but it remains true regardless.
You could say the available information provides evidence that gun ownership does not cause increased crime, which would be true. It is however very much possible that it does increase violent crime, and something else is responsible for the decline.
If you need a list of possible influences we can go with better education, reduced environmental contaminants (someone mentioned lead specifically), the rise of easy access to information (the internet), deterrent through harsher penalties (which I do not personally believe helps), economic conditions, etc.
There are rather a lot of things which would influence the crime rate and are not accounted for in a simple crime vs gun ownership chart. In this case nobody will be proving or disproving it either way, it is simply too messy to be subject to this kind of analysis.
The reason so many people are coming down on Jane's post is that it redirects the argument in a very misguided (or underhanded) way. A logical process is given with an example where it works correctly, then applied to a situation where it does not.