There are a few things in there that made me raise an eyebrow. Humans don't really experience much neurogenesis. There are some areas where new neurons can form, under certain conditions, but they tend to be special purpose ones, and the older structures in the brain as well. The thing that really differentiates us from other animals is our overdeveloped cortex, particularly the frontal lobes, but the neurogenesis that's been found is mostly in the deep gray matter and is more associated with things like motor coordination and reward processing. One interesting exception is the hippocampus which is known to be important in memory formation. Indirect hints of neurogenesis in the cortex have been reported, but other methods that should turn them up haven't, so the evidence is contradictory. I'm also not aware of neurogenesis being particularly pronounced in humans. It occurs in other primates, and in other vertebrates.
There does seem to be a connection between intelligence and the brain to body size ratio. Bigger bodies require more neurons to carry and process sensory and motor information, and these neurons are probably not involved in intelligence.
What we call intelligence seems to me to be likely an emergent property of a bunch of neurons that don't have any pressing sensory or motor tasks keeping them busy. Various factors affecting communication efficiency and interconnection among neurons are probably important, but these can be disrupted quite a bit in human disease and the sufferers don't lose their human intelligence (although their cognitive abilities do decline). I don't think there's a magic humans-have-it-and-nobody-else-does bullet. Human intelligence is just what lots of animals have with lots of extra capacity, possibly redirection from other things (like senses) to boost that capacity, and maybe a few tweaks for optimizing neurons that talk to themselves over ones that communicate with the body.