I’m going to pick apart several of the things wrong with this statement.
First, the early U.S. economy was not dependant on slave labor, in either the future Free or Slave states. There were a lot of them, but did not form a key or distinct sector. Nor was the nature of slavery fixed as firmly as it would two generations later; at the time, there was little difference betwee a poor white laborer, an indentured servant, and a slave.
Second, the population of black slaves at the time was much, much smaller; it grew very rapidly post-war.
Third, it wasn’t very productive. It didn’t become really profitable until the introduction of cotton cultivation alongside the rise of textiles in England, both partly a product of the Transportation Revolution. It wasn’t really until 1830 that cotton exploded, and the handsome “Old South” was a fairly recent innovation at the time of the Civil War. (Strictly speaking, the boom began much earlier, but needed fast, cheap water and rail transportation to fulfill its potential it also wasn’t identified with the slave plantation economy early on).
http://www.slaveryinamerica.org/history/hs_es_cotton.htm
The raw growth of cotton might seem to be a smooth growth - or even growing faster in the early years of the Republic. However, you’ll note that the growth of cotton rose even after the price dropped immensely. While the deflationary nature of gold and silver might account for some of this, it’s not nearly enough for all. Cotton was profitable only with a huge international market on that scale.
Fourth, slavery was probably correctly viewed as being on the way out in 1800, precisely because big plantations weren’t really all that profitable. Some families managed to hold large fortunes, but these were mostly large landowners in Virginia. Tobacco and sugar simply couldn’t guarrantee a slave economy.
Fifth, the Free States were more stable and wealthier than the Slave states, and grew to be even more so, combining industry, commerce, and agriculture. This is altered a bit because the wealthiest counties in 1860 were in the slave states, but these were simply big centers of the cotton trade, which combined low population with lots of cash flow. For a modern equivalent, think of Hollywood.