I disagree with Dawkins. If genes were the unit for evolution, we wouldn't exists. We would all just be bacteria spewing out copies of genes, with the planet buried in a pile of tiny protein chains. Genes are the servants of the species. They are a tool used by species to pass on traits, or even to acquire traits from other species. Genes are inert absent a species using them.
that suggests that once born, an individual could have all of their genes removed, and be perfectly fine, other than not being able to pass on traits to a new individual.
Yes. Genes and DNA are not the same thing. DNA is a means of recording and passing along genes. And alien organism that doesn't use our system of DNA/RNA could still possess genes. So it is theoretically possible to remove all manifestations of genes and have the organism continue to function.
This is not very far fetched. Much of our DNA is used for a short time during development and then effectively turns off for the rest of our life. Taking that DNA away might go unnoticed.