My impression as an atheist is that there is no special sauce that makes human intelligence impossible to achieve. It will happen eventually.
Our brains are computers made of meat. Nothing more. Our thoughts, our dreams, our consciousness itself is quite literally just chemicals, hormones and synapses instead of circuits, binary code and wiring. There is no soul that would prevent true life from arising once the computing becomes powerful enough for it.
The first two thirds of Seveneves is really good at exactly what you describe. Once you get to the third part (you'll recognize it) just pretend the book ended before that.