Whether income inequality is the result of a fair process is not the main question that needs answering. Is it fair that I was born with a usefully high IQ into a stable society that craves knowledge workers? No, it is a privilege and has nothing to do with fairness. Yet it helps no one to take that particular privilege away.
Instead, we should accept that some will be privileged and some will be disadvantaged, strive to improve living standards for everyone, and lift up the less lucky by way of redistribution and public services.