I don't think their statement implies that. Note that they said it will be the end of the American success story, not the end of the American
immigrant success story.
The nature of the American success story changes over time and with that the nature of immigrant success also changes.
In the last decade or so tech, especially information tech, has been one of the biggest contributors to growth in the US economy, and first generation immigrants have been a big contributor to that. For example, first generation immigrants have founded many of the tech unicorns (although I think he overstated it a little--my searching suggests it is closed to 40-50% rather than a majority).
In earlier decades the biggest contributors at various times included manufacturing, farm technology, defense, the Gulf Coast petroleum industry, and construction.
There were certainly immigrants involved in all those but not nearly to the extent that they are in present day tech, especially at the top.