When used in the argument in favor of gun control, the argument divides the world in a very black and white scenario: either a gun kills someone, or a person kills someone, with no gray scenarios in between. Any real life situation is not black and white, it's a complex situation full of preconditions and probabilities.
So for the argument to be used in the situation of gun control, we'd need to know what is more dangerous overall:
1) a world with guns
2) a world without guns
and of course, this is also a black and white scenario, but we'd have to view these as two ends of a spectrum of gun control (no control vs absolute control).
Then we would have to decide whether it is easier in general to kill with the aid of a gun, or without the aid of a gun. We would have to oppose that also to using another inanimate aid regularly available. E.g. if it is equally easy to kill someone with a hammer, then hammers could be equated with guns, and they should follow all the same rules.
Put in a formula we would calculate
P(fatalities) = P(availability) * P(effectiveness) * P(motive)
This could be done for any object, whether inanimate or animate. Of course objects don't have motive, but we could say that a tsunami e.g. has 100% motive to kill you, since it is aimed at you and cannot be stopped or avoided easily. It is also very effective, but luckily not very available.
I think the biggest flaw is the implied 'person without a gun'. Since we're used to saying something like this in an 'exclusive or' context, e.g. "I am not blind, I am deaf". In that sentence you're implicitly saying that blind people cannot be deaf. So it can be rephrased as A)"guns don't kill people, people without guns kill people". Unless it is meant as B) "guns don't kill people, people kill people with the use of guns". So the first thing in the argument is to ask the person which version they mean to communicate.
A) prove that it's easier to kill people with the use of a gun than a non-regulated item, such as a hammer
B) guns are still part of both equations, so by taking out guns we take out the loss of lives. Think about gun accidents, where people clean their gun and it accidentally kills their wife. Or a child playing with a gun, and it accidentally goes off. Or a bullet ricochets and hits an innocent bystander. So in all those cases, it wasn't the intention of the person to kill, but it still happened because of the existence of guns.
To just take it one step further, and go completely reductio ad absurdum: if people kill people, then we should initiate people control instead of gun control in order to prevent killing. I think that's way more scary!!!