This is a very rough verbal sketching of some of my thoughts concerning the points below. Basically, I don’t think that our desire to mate and reproduce is solely responsible for both the growth of our brain size, and the reason for all of culture and its achievements. Rather I believe our human uniqueness in culture, language, and brain size emerged from a multitude of factors, which produced a result which cannot be reduced to any particular aspect of animal nature.
How can culture be a produce of desire when most of culture is done in the absence of it?
Humans are like an invasive species because of our ability to (over)adapt.
If desire is central to human uniqueness and intelligence, then why is it not more common in animals?
A piece of our animal nature I believe, cannot be responsible for our human uniqueness. It was most likely a mutilated of factors. These factors may have made a whole greater then the sum of its parts (Gestalt) and that sum is our uniqueness (language and culture).
This may have been a kind of strong emergence (like the big bang from nothing, or mind from neurons and brain-world).
The fact that culture and language are both transcendental and referential makes it beyond any animal necessity.