I guess it all depends on what your definition of the American Dream is. It's all relative.
Definition 1 - In America, it doesn't matter who you are, you can be successful. You could run your own business or attain a high-level political office (to name a few).
Definition 2 - In America, if you work hard, you will always attain a high level of success. You will never be treated unfairly or have to obstacles you need to overcome in order to succeed.
I think if you ask most immigrants who came to the US from countries with authoritarian regimes, they'll agree the "American Dream" still exists. Of course their perspective is quite different. In their home country no matter how hard they worked they could never be successful because they were the wrong race, member of the wrong political party, didn't have connections to the right people, didn't go to the right schools, etc.
I don't think any reasonable definition of the "American Dream" demands that America is perfect. It's just an observation that America is very unique in the world in the opportunities it offers people from all backgrounds. Heck, I've spoke to European friends who feel that the US offers more opportunities for everyone than their home country does.
I'm not sure what your point is. Are you agreeing that the "American Dream" is now (to some extent, which may have been different in the past) part of the effort of convincing people that the game isn't rigged against them, or are you disagreeing with that?
I think that this almost indestructible believe that anything is possible is a central part of the US existence. Back then, when somebody wrote “all men are created equal”, this statement could be already seen as an insult to large parts of society. The reality is, if people have high hopes and dreams, they are most likely not going to achieve them.
But I am absolutely certain that these give them a purpose to try harder and become better in whatever they do. Even so it is largely an illusion, it benefits society overall as it motivates people to live up to their best possible capabilities.
But I don't believe that there is anything such as a fair game and that those not benefiting from the current rules of the game will always call it unfair. This is similar to the discussion in the article of the Ivy Schools. Is what they do fair? Absolutely not! Does it serve its purpose of positioning the schools as institutions that bring together the intellectual and societal elites? Yes it does.