>Previous studies have showed that owning a gun increases the odds of injury by gun
No they haven't. The studies show that people who own guns have higher likelihood of being injured. It does not show that owning a gun /increases/ odds of injury.
Places that have wet streets have a higher likelihood of it raining recently. Does that mean that wet streets cause rain?
You are assuming that the non-gun-owning population and the gun-owning population are the same. This is illogical. Perhaps it's the case that people who buy guys are just generally risk takers/more reckless than the general population (which is reasonable given that they're purchasing a gun in the first place). The problem with your logic is that its vulnerable to Simpson's Paradox[1]
My arguments are not broken at all. To do the type of control you suggested above would mean putting guns in homes of people who would otherwise not have one. Previous studies have showed that owning a gun increases the odds of injury by gun. Putting subjects at that sort of risk for a perfect control study absolutely has ethical implications.