The core concept of capitalism is that capital works for you. It separates earning money from working. If you have enough starting capital, you don’t have to work at all, because you can hire people to work for you and you can even hire people to do the hiring for you. You can even abstract all that away by not hiring but instead investing, and you can even pay someone to do that for you.
If you have money in a capitalist society, you don’t need to work, and if you don’t have money, you can work all you want and you will still not become rich.
On the other hand, the American Dream says that if you just work hard enough, you will be successful, you will make it, you will become rich, no matter whether you have money and contacts or not.
These two ideas directly conflict each other.
No they don’t conflict; they’re symbiotic.
The “American dream” was dreamed up by rich capitalists to encourage poorer people to work harder, longer hours, and to do so with as little grumbling as possible. Couple it with the “rite of passage” mentality - i.e. working super shitty conditions because everyone else had to do the same - then it’s the perfect indoctrination needed to convince people that their hardships are their own failures and not the system’s.
Another, simpler, way to look at it is that the “American Dream” is a carrot on a very long stick that is dangled in front of us to encourage us to move forward with their capitalist agenda while making us think we’re getting a reward for our labor.