That's nonsense. Most of us southerners have ancestors who did their own farm work. Only a small perecentage of southerners owned slaves. A complaint of small farmers was that they literally had to "compete against slave labor."
It's true the south would not have joined the US without legal slavery, but the reasons were not that it was "totally reliant on slave labor". Probably just the big money interests and their backers were near totally reliant.
While most farmers in the south did not own slaves, the statement "the economy of the southern states was... reliant on slave labor" is a statement of fact.
Most of the farms in the south wwere engaged in something approaching subsistence agriculture, their surplus production constituting only a small proportion of the entire market for a given commodity.
It was the larger estates and plantations -- who were reliant upon slaves -- who produced the bulk of the trade goods and export commodities. Ergo, they were the linchpins of the "economy".