Luckily, I think we can mostly go a long way towards testing this assumption by looking at immigration. We can, for instance, look at demographic groups from developed countries where guns are not readily available (Maybe Japan, Australia, China, UK, or others) and see how these groups crime rates in their country compares with immigrants to this country surrounded by easily accessible guns.
If the author’s thesis is correct, then there should be a noticeable difference once these groups are exposed to easily accessible guns.
Is this the case?
That doesn't seem to track logically, to me. You're assuming that the assertion is only useful if those are the only things that cause gun violence, rather than the primary and addressable things. Of course it's a myriad of things of varying impact and varying interaction with each other. This test can't realistically control for those things, nor can it even control for the basic effects of transitioning between two countries/cultures that would create noise in the data. This test could certainly produce useful information, but it's too naive to prove or disprove the notion that availability and laws contribute strongly to US gun homicide rates.