I believe that it has been suggested to you before but let me also encourage you to read The Ancestors Tale by Dawkins. In particular I recommend The Peacocks Tale where he lays out his theories for why the human brain is the way it is.

One of the best things about this book and Dawkins in particular is he is very honest and upfront about what science knows and can know. On this issue there is no consensus or even widely accepted position. Dawkins lays out a convincing case that our brain power is largely the result of sexual selection. I cant do it justice here and wont attempt to but like I said I recommend the book.

One other interesting point that Dawkins makes is very realted to your lion v prey scenerio of co-evolutionary harmony. There are recuring themes and patterns in evolution, prey and predator each get faster, bigger teeth for lions and predators result in thicker skins for prey, etc. When you find something in nature that is unique theres a good chance that sexual selection played a big role. Sexual selection kind of sends species shooting down evolutionary paths that are not expected and dont necassarily serve any survival purposes.

All that being said I dont know if I even addressed your main point. I dont know if I would agree with the theists but I do think there is something other than just increased intelligence that seperates us from animals. The soul and related concepts I dont think anyone has a good answer or explanation for. However, I do think that Dawkins and materialists like him are much closer to the truth than any theist. I suspect that whatever power of awareness that we have is also present in animals to some lesser degree.