K&L Gates is in the same building as the company I work for, despite multiple mergers 2 decades past retirement the Gates name is still on the door and it's not because of Bill.
William Gates is a very accomplished individual and a true philanthropist not many people would be schlepping through Africa in their 80's and 90's, especially people of his stature.
1976 (when Bill was 20) would be about 10 years after W. Gates co-founded his law firm and while for sure that they were well off that time I don't exactly know if they would be classified as "wealthy" especially by SV standards.
https://en.wikipedia.org/wiki/William_H._Gates_Sr.
My take on this is that if anything this proof that good parenting is important, there are plenty of other wealthy lines where the children do not amount to anything but managing not squander their heritage and fortune.
Don't get me wrong growing up in an upper middle class family or higher does have a lot of benefits it allows you to have the potential benefit of better education and values and it allows you to take risks that less fortunate individuals may not be able.
However I don't see anything that would prove that someone's success will change their children's future other than to give them a bigger bank account (which cannot be dismissed on it's own ofc).