Here we go again...
When I read this, in my head, this analogy comes to mind: "We could stop bombings by allowing all citizens to carry concealed bombs [to use in self-defense]."
Now obviously that is an absurd analogy, but it mirrors a lot of what you're suggesting: fight a dangerous weapon with another dangerous weapon, potentially put bystanders in harm's way, make it even harder to track/identify the good from bad, make it easier to obtain weapons.
The US has more mass killings than any other western nation, yet they also have concealed carry, clearly something isn't working here. Is the solution really "more guns, more guns?"