I think this predisposition may be not for equality but for cooperation. We have empathy, we have child care hard-wired and we have some sort of protection against cheaters, to benefit the whole community. (it is not a cultural invention, it is, like you've said, genetic predisposition).

Equality/inequality is, I think, a cultural factor. Culture is also subject to some sort of evolution process, which means that some inventions got lost, but some remain and evolve. So, more equal society is indeed more stable and healthy. You might not see so intense struggling or competition which means less innovation and so-called progress, but innovations or progress aren't among ultimate goals of the process of human evolution.