100 years ago people would have thought mental arithmetic, memorization and rhetorical speech were core parts or indicators of intelligence. I think that most people today would see creative problem solving, social empathy and long term planning as more indicative. In fact the 19th century indicators are now seen (in popular culture) as part of a disorder (autistic spectrum) that is often used to explain the behaviour of otherwise incapable people.
Our culture, communities and technology define our humanity, creating a greater humanity through breeding misses the point somewhat.
A baby monitor with progress notes on various pivot options sounds great. (Revolutionary Girl Utena model Serta...) It beeps if all the child's career options are deprecated by the next 15 years, and points to an appropriate hill near which those careers fester; or if their social health role is to force vulnerable mutations in (otherwise) epidemic disease (I'm only a Doctor ironically, etc.)
Of course this does not mean eugenics is a desirable activity that we should be engaged in, but I have not seen any rational argument that it has been “proved" to be impossible.
Also I don't think it's agreed.
Also it's not clear that your statement that population level changes can be created by selective breeding is true - if a characteristic is dictated by alleles that damage fertility or other fitness indicators then it may well not be continuous. In the case of human intelligence the increased child birth mortality associated with large heads and the increase dependence of infants on parents are examples. There may be others which are more relevant to modern life, like propensity to depression, disinterest in sex etc (note : I am using these as examples, not saying that this is so).
Of course selecting only for GCA may well increase the frequency of other phenotypes, but unless GCA is directly linked to that phenotype then there is no reason you can’t select against any undesired phenotype at the same time. This is done all the time with the selective breeding of animals and plants. I know of no phenotype shared by all individuals with a high GCA so there should be no phenotype that would inherently be a consequence of selecting for high GCA, certainly not infertility.
Use whole-genome mutagenesis and increase the size of your population. This will reduce the total number of generations that you require to find the novel mutations that you seek.
I don't buy that argument. There's no reason to think that evolution has been selecting for cognitive ability as strongly as it possibly could.