An emergent property is a property that arises in a structure as complexity increases. Consciousness is indeed a property of the brain, or more likely, a property of the brain-body complex. As far as we know, insects and reptiles are not self-aware, though they are aware of their surroundings and react to them. But once the brain became sufficiently complex, somewhere in our evolutionary history, consciousness emerged. You are probably right that it did not suddenly appear. It emerged gradually. just as the eye of an eagle did not appear suddenly, fully formed: It evolved over aeons.
I am quite skeptical of the list of "famous" predictions of the past in the link you quote, though it is true that the more we learn about the world the more we are able to manipulate it. However, listing things people once thought were impossible as an argument that any particular thing will be possible in the future, is a logical fallacy. Just because they laughed at Fulton does not mean that one day we will have transporter beams. Just because some famous person thought human flight was impossible, does not mean that one day we'll have self-aware machines. We might, but that argument is not valid.