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.
The American Dream is inherently capitalist, it being a myth doesn’t change that.
The crux of the American Dream is that you have to suffer on the bottom of the totem pole, but eventually you’ll get the chance to be on top and exploit the others on the bottom. The American Dream is very useful to the capitalist class because it gives people motivation to stay in the rat race, to believe that they have a stake in capitalism as a system, because one day their hard work will be rewarded and they will be a capitalist as well.
Outside of the context of capitalism, the American Dream doesn’t really make sense. If realizing that it’s a lie helps push people to the left, that’s good and should be encouraged, but I don’t think that makes the Dream itself anticapitalist.
The purpose of the dream/myth is capitalistic, but if you analize its contents like OP you end with ironic conclusions. You are both right.