You're underestimating the power of connections. I would never say that hard work won't make you successful in the US, but coming from a wealthy family creates a lot of opportunities earlier on in life that poor kids will have to work hard to see. Coming from a wealthy, or even upper middle class, family means maybe your summer senior year of high school is spent interning with your uncle's firm instead of working for extra spending money. It means you can ask your parents for career advice (poor parents don't have career advice because they don't have careers). It means you have an idea, from growing up with it, what success in a career path in a developed country looks like. All of these things are tremendously valuable.
Passing down of beliefs and behaviors is almost more valuable than passing down money.
You're embedding positive or negative behaviors into someone's personality.
1>Passing down the belief that education is valuable
2>Passing down the belief that investing in the future is more valuable than short term spending.
etc.
Also having resources in general is a massive benefit.
1>having security allows wealthy kids to take more risks/reap more rewards.
2>More free time to learn and advance themselves
3>ability to take low paying jobs purely to make connections/build job skills
4>Status is a language that every one understands at a primitive level and so just getting more respect in general which means better promotions, better mates, etc.