I do not believe it was staged. While there is a risk of danger, I believe it is a small risk. Perhaps a the axe head flies off, or a block of wood shoots out and hits the kid. But those risks are small, and it doesn't make a lot of sense to spare a child the experience of physically working to heat your dwelling in order to insure they are safe from an unlikely accident.
Anecdotally, my dad and I drove into the woods and cut down trees for the fireplace in my family's house. After he cut down the tree, he would cut the tree into 1 foot rounds and then split them with a maul. The cut, triangular pieces had to be loaded and stacked into the back of a pickup, and it was back breaking work.
However, every time the work was done, I remember feeling a sense of accomplishment and contemplating how it was the drudgery that ultimately created that feeling.
I think that we should not spare our children these enormously valuable learning experiences in the name of safety.