If number of children born were spread more evenly among all parents, would our species' genetics have more variability?
Well the obvious answer is of course yes of course it would. That is the essence of natural selection. Variability leads to different phenotypes and selection for and the amplification of beneficial traits. Evolution. Except if there is no selection (everyone has the same amount of kids) no traits are amplified so everything in a large population is static. Until you introduce mutations and that increases variability. Eventually each individual would have such a large degree of variation from the next that I would assume speciation would be a very imminent result of any form of seperation or selection, which is against your original premise.