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.
American Capitalism, with its abundance of neoliberalism, works on the premise that given no external involvement, the market will take care of itself. Companies will make products that people like, or will be out competed.
The American dream extends from that idea that workers can work wherever they want and have completely free movement. So if someone is smart or a hard worker, they have plenty of opportunities and will eventually be successful.
In reality, neither of the above are true. Markets are not capable of taking care of themselves, e.g., because there is inertia, inelasticity, and barrier to entry for many high-capital businesses. Government has propped up most “desirable” large industries through heavy subsidies and tax breaks, like oil, farming, telecom, and tech. Workers do not have free movement, some from self inertia to want to stay close to roots, family, friends, but mostly from the same neoliberal policies that remove social safety nets and fail to provide everyone the basic necessities.
Add to that the fact that a solely money based capitalist system has no ability to measure environmental degradation, wealth inequality, or population satisfaction. And the government is more than happy to step in when it’s businesses being hurt vs people - think too big to fail or propping up businesses during covid/after natural disasters.
Even if a true capitalist system were allowed to exist, it is ultimately anti-competitive. A business in a segment that’s doing well will slowly acquire other businesses in the segment to become a monopoly. Eventually the monopoly will keep growing and acquire the largest businesses in other segments. Besides regulations, technology disruptions can break this cycle but those are fairly rare, and are mostly a recent and likely short lived phenomenon.
Finally, capitalism requires economy and population to keep growing, in the absence of which, there is complete stagnation of movement and the system will collapse into feudalism.
Anyway, I think you are both right and wrong. Capitalism as people imagine it to be feeds into the ideal of the American Dream. But both true capitalism and its reality actively thwart it, by closely interlinking the economic system with the political.