No, I disagree. Sorry. It's not proof of anything but the number of kids who died accidentally of gunshot wounds (and even that tends to get underestimated because of classification issues, according to links that have been quoted earlier).
The children who die of accidental gunshot wounds are just a fraction of all the irresponsible adults.
Let's not forget the cases where the kids didn't die but got injured.
Then there are the cases where children didn't die themselves but they killed somebody accidentally. If it's an adult victim it would be invisible when we just look at children's mortality statistics. One of the links that I posted earlier said that it's difficult to get information based on the age of the accidental shooter so there might not be accurate figures about this. Maybe someone can find them.
Then the kids who shot and injured someone accidentally but not fatally.
Then there are the cases where kids got their hands on a weapon and shot somebody on purpose. They'd be classified as homicides or homicide attempts but there's still some irresponsible adult somewhere who allowed them to have access to a gun.
Then the cases of children's shooting suicides. Again, not classified as accidents but nevertheless some idiot allowed a suicidal child access to firearms.
Then there are the undetermined cases of children dying of gunshot wounds, some of which might be relevant if known.
Then there are the kids who got their hands on a gun but couldn't or wouldn't fire it, who shot an animal, who shot harmlessly in the air or caused a bullethole in the wall, lost interest, or an adult noticed the danger in time to take it away before anything could happen. Most of these wouldn't be reported and we won't see them in the statistics because no one was injured.
Then there are the kids whose families get consistently lucky and avoid incidents although they're store loaded firearms openly where the kids' friends could find them, even if the family's own children might know better. Again, not coming up in the mortality statistics, but I quoted a survey earlier where 85% of gun owner parents said they didn't store theirs safely although they thought it was important and I've seen several posters here on WS indicate that in their family weapons were never kept away from the kids so I'm thinking there are probably quite a few of these. The thing is, you won't know if anyone is going to get hurt until after everybody's grown up, so I don't necessarily think these people were any less reckless just because they got lucky and ever nothing happened.
Someone asked about gun storage in families with kids earlier and here are some numbers:
http://smartgunlaws.org/guns-in-the-homesafe-storage-statistics/
http://pediatrics.aappublications.org/content/early/2014/01/22/peds.2013-1809.full.pdf+html
In 2009 there were 7391 hospitalizations for firearms injury in children and adolescents. 2149 of these were classified as unintentional, 413 as undetermined, 270 as suicides. There are even two suicides in the age group 0-4 years and 2 in the age group 5-9 years.
I don't know...two may be a vanishingly small number but IMO there should be exactly zero cases where a child that age shoots themselves to death on purpose.