However, it does not follow that if the genome is not very complex, the brain must not be very complex, because the brain may be built not just based on the genome, but also based on information from the outside environment.

I think this is relevant if the topic is uploading, but I think it misses the point if the topic is "how hard is it to produce a self-improving intelligence". The brain of a newborn has not yet received much input from the outside world, yet has the ability to learn. This places a (rather large, IMO) upper bound on how much complexity is necessary to produce an intelligent system.