On his Reddit ama he himself said that his upbringing contributed a lot to his success.
Also this doesn't say more about US but more about confidence and ' smoke and mirrors show' while starting out.
Don't get me wrong no one is arguing that money doesn't affect your future of course it does but it's only one factor and a factor with a lot of diminishing returns.
I don't personally see that Bill got where he did due to some exceptional wealth both because I don't consider the Gates pre-Microsoft exceptionally wealthy by US standards and that wealth can only grant you opportunities.
Considering who his parents were in terms of their achievements I'm also not so sure we can claim that Bill Gates without Lakeside would've not have been 'Bill Gates' sure it's nearly impossible to estimate the effect of each individual moment in ones life but if we take it on a macro level I think the character of his parents influenced him considerably more than their wealth.
To make my point even clearer at some point say magically $250,000 a year I don't think wealth is a factor in your success unless you completely inherit it (e.g. inheriting a business). At the point when you can live in a good and stable home, receive good education and discipline both at school and at home money isn't going to be the deciding factor in your success if it's your success.
I think that may say more about the US than it goes about proving your point.
People greatly over estimate the level of power and influence the Gates had when MSFT was formed, everything I've read about them shows that both Bills parents were remarkably accomplished people.
When you think about old money and the rich and powerful organizing a divestment campaign against the apartheid in SA like his mother did isn't the first thing that comes to your mind.
Out of all of the examples of "Wealth begets Wealth" the Gates are probably the worst one I can think off.