I think I had read the argument that the complexity of the human genome is a upper bound to the "innate" part of the complexity of the human brain before (either somewhere on Language Log or in Motion Mountain by Christoph Schiller, IIRC).

(Of course, this assumes a narrower definition of innate than yours, because "the overwhelming majority of the conditions animals of a given species actually develop under" share lots of complexity. In particular, according to that definition linguistic universals are innate by definition, whether or not it would be possible in principle for a population of genetically normal humans to adopt a language that violates them.)