Yes, being rich can make you confident which can do well for your career. But there are plenty of other unfair advantages that also bring confidence, like height or good looks. And then there are really horribly unfair things like intelligence.
On the other hand money is concrete and real. If I had 5 million in the bank my attitude would be 100% different. I wouldn't be considered geeky, I'd be geek-chic and eccentric. I wouldn't have most of the anxities I have now because I know money can buy me almost everything I need.
Sorry, but large sums of money are nowhere comparable to less tangible gifts like height or looks. One buys me a home the other maybe gets me an occasional second glance.
I'll also toss in that thinking of yourself as a social peer with your bosses is a huge benefit. The book "Outliers" explores this. Well-off kids are raised to be assertive and have a sense of entitlement. They just do better in life. They can defend these attitudes by saying "I come from a well off family, just as wealthy as you. I went to the same school as you and your kids and my parents know the same people. I was taught to get my way and that my parents will protect me with their riches."
Lately, coincidentally, I have a lot in common with my current bosses. The social parts take care of themselves and its a huge benefit without me putting in a lot of effort. I don't feel the divide between them and me as much as at my previous jobs. I have a small amount of savings that can help me get by if I get fired or laid off. If this is how rich kids feel at age 23 when entering the workforce then they have a HUGE advantage.