Knowing how to properly use something and knowing when its use is "appropriate" are two very, very different things.
I don't think it would be hard to teach a six-year-old either in this case. Seems pretty simple.
Six-year-olds get scared more often doesn't translate to significantly greater chance of a specific scenario transpiring where six-year-old with ownership of firearm gets scared and shoots someone who was not a threat.
Personally, I would never say no to a six-year-old who wants to have access to means of self-preservation in a worst-case scenario. I also wouldn't presume she hasn't been appropriately educated for her own and others' safety. I'm quite cool with this.