I'm sure De Waal addresses this in more detail in his book, but the omission is perhaps less glaring and capable of destroying the entire argument as you present it to be.

Although it is clear that human language is unique and appears much more developed than any animal communication system observed in nature, no one knows what exactly property of language makes it unique. Indeed, infinite use of finite means, or syntactic structure are often put forward as candidates but syntax and generativity have also been observed in bird song. We simply don't know what the defining feature is that sets language apart from animal communication systems.

Similarly, we can't identify the unique cognitive ability that differentiates us from other animals. Since, as De Waal points out, humans are genetically speaking very close to our nearest sibling species, and we simply don't understand what precisely makes us cognitively different from, say Bonobos, the standard assumption should not be that we're somehow gifted with very special cognitive abilities, but that animal cognition is much more similar to our own cognition than is often assumed. Rather than studying where animal cognition, especially in genetically related species, resembles human cognition, we should be studying where it differs from human cognition.

Keep in mind that anatomically modern humans have only been around for two hundred thousand years, and the explosion of human culture only happened around ten thousand years ago. Not a lot of time for massive evolutionary changes.