I think there are a few factors unique in U.S. history that would naturally lead its people to believe that hard work alone is sufficient.
Firstly, I think times of change are the catalyst needed for mobility (up and down). Just like in organizations: fast growing companies are where many employees first get elevated to, say, the CTO or Director level, and which they can then maintain from company to company. They worked hard, sure, but not harder than anyone else. It was that disruption in the status quo that gave them the opportunity in the shifting sands to find elevated ground.
I think the other factor is points of leverage. A sibling comment points out slavery, colonialism, cheap labor. Those are general points of leverage, but there are others (and possibly not the most important ones for the U.S.)
So the first great change in my mind was just the establishment of the U.S. For decades we pushed west "discovering"/stealing new territories and its natural resources. This led to mobility for some, but the leverage is small (still need lots of people to work your massive land stake in Kansas). Still, those resources were sent back to Boston or wherever and led to many opportunities.
Second, the Industrial Revolution. Now you're getting into great leverage possibilities. Vanderbilt was an ace accountant, but what other time in history could somebody good with a ledger rise to the richest man in the world? Vanderbilt was an exception, no doubt, but there were plenty of people who had accounting skills who significantly improved their station in life (and hence their subsequent generations).
Next was WW2. With Europe ravaged by war, the U.S. naturally was able to rise and fill various roles previously filled by the British and others. (Maybe the most important one here is reserve currency status.) This created ridiculous opportunity for its citizens to make a name for themselves. This is the wave many boomers rode to their current levels of wealth.
Lastly is the computer revolution. The wealth from this one has probably been more concentrated, but the leverage possible has improved the station of many many people (myself included).
So I think hard-work (or related things like "savviness") is necessary, but not sufficient. Maybe now people are just coming to terms with a world / country that is mostly settled, and without that disruption, the chances of moving up in life are greatly diminished back to where they've been for most of human history.