I think that instead of framing it as fairness--which seems to agitate those who were born with monetary advantages, this conversation should be re-framed to focus on societal efficiency.
It is inefficient to shoehorn people to the top because their parents were at the top. The people at the top should be our highest achievers and not our decent achievers who were born with rich parents.
The way our society is structured we will find ourselves much better off in a few generations if we can devise a system where each persons potential is given a proper environment to flourish and where each high potential person is given a carrot to chase that isn't unachievable.
Edit:
Speaking of carrots to chase. It's looking more and more like the wealth of millennials(and I suppose zoomers?) long term will be very very determined by their inheritance(if they are so lucky to have one lol) and less so by their hard work in life.
This is likely the cause of the number of people under 40 who seem to be "opting out". It's very difficult to get to the supposed goal unless you are a high achiever. The carrot is very much too far away and this is another societal inefficiency. We would have a much more dedicated workforce if we made it possible to get "carrots".