Nothing has changed. It is still as it has always been.
The US space program and the Moon landings can be traced directly back to Wernher Magnus Maximilian Freiherr von Braun and his work at Peenemünde, utilizing slave labor to build rockets for Nazi Germany.
"More people died building the V-2 rockets than were killed by it as a weapon"
The course of history was shaped by Von Braun being terrified of the Soviets and choosing to surrender to Americans.
Some people see this sort of historical note as extremely relevant to everything today, and some don't. I see nobody trying to bridge that gap.
The traditional american dream is about social upwards mobility from rags to riches, about a promise that all that is needed to become wealthy is hard work, determination, a bit of ingenuity and puritan virtues. In its classic version it often features an immigrant family who comes to america with nothing but worn out clothes, works hard jobs and long hours to create a livelihood, overcomes hardships and struggles so their kids can get an education and better jobs. It often also calls out oppression, poverty and war in the old world as reasons for migration and to further highlight the ideals of liberal society in america.
The modern version ignored the generational aspect and instead focused on individuals becoming millionaires. Entertainment loves extremes, the lower the start and higher the end, the better. The quicker the journey, the better. And this became accelerated to the absurd. Be poor, wash dishes, millionaire. Since this is so very unlikely, and real people who fit it, like Rockefeller, became so unpresentable, media turned more and more towards fictitious comedies. Yet this version is still very dominant in culture, because it calls out a truth: when asked what the american dream is, many will say it is about becoming very rich. That is, what the modern american dreams of.
Now in contemporary media we see a return to the original idea, but with twist lampshading its problems. Migration is no longer a central topic, america no longer the beacon of hope for the tired, poor, huddled masses. Instead we see a focus on the disenfranchised and marginalized within american society. The post-modern MTV generations version of the american dream is B-Rabbit living in a trailer park with his alcoholic mom, making it as a rapper despite his peers attacking him for having the wrong skin color. It plays the core straight about making it with hard work and being virtuous, but it doesn't only look at materialism and individualism, as it also expects society to hold its promise of being liberal. The postmodern loves to play with dualities and reversing structures.
Lastly the post-modern satire version instead picks up the part that got dropped, the dream of becoming very rich. It teasers a protagonist from an african minority and reveals a wealthy and well educated white man from south africa. It shows upwards social movement from upper class to ultra rich, subverting the idealistic to instead highlight the reality that those who start well have more opportunities to venture and do even better. It holds up "being born rich" like Diogenes holds a plucked chicken, and calls out "behold: the american dream"
Is that how some people conceive of the "American dream"? I don't know, but I think it's the easiest to find factual evidence of happening over a long period of time.
-- a idk probably 10th generation American living on the East Coast.
Neither your modest scenario nor becoming a millionaire are specific to the US. And neither can possibly happen for all Americans.
So if that's really what the American dream is defined as, I think it inherently defines the US as a failure.