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.