He was in the enclosure for ten minutes, he wasn't dragged around for ten minutes. The official police report suggests between two and three minutes "within close range" of the gorilla.
Once the gorilla reacted to him violently, they had no choice but to shoot. Sedating a gorilla isn't quick and there was too much of a risk that the dart would richochet and hit the boy, or the darter would miss. The level of sedative needed to quickly sedate a large male gorilla would usually be fatal to a four year old child.
The police report says the child appears to have non life threatening injuries.
Im sad for the gorilla, and the boy. There were a lot of failings here, and they both suffered.