Is it fair to say it’s a general human trait to iteratively, over many generations, step towards that society?
It’s difficult to say, because there are plenty of fairly horrible examples where an entire nation gets trapped in a culture that actively masters over generations how to punish honesty and reward corruption. Russia being an example.
misanthropic programmers say no
Cooperation wins in the end, so I'd guess as much
Nope
One might almost say evolutionary. https://pubmed.ncbi.nlm.nih.gov/31163155/ , for example