I'm not sure what's so hard to believe about that. It's not even a controversial statement. Are you suggesting that if labor were cheaper the market would not consume more of it?
If there's a demand for X amount of product, and it will take Y people to provide the labor to supply that product, then businesses will hire Y people. They're not going to hire more than Y just because they can.
The only time the market will consume more labor because its cheaper is if the amount of labor they already had was already insufficient, in which case if they really needed more people, they would have already hired them.
That's just wrong and demonstrates a fundamental lack of understanding of basic economics.
If labor were much cheaper, I personally might be willing to hire a maid, a gardener, a masseuse, a cook, etc. Similarly a retail business might hire more cashiers, or have more staff on the floor to answer customers questions and help them find things. A gas station might provide a full service experience for their customers. You could do this analysis for virtually any type of business and given cheaper labor they would hire more people.
Why is this so? Because the demand for labor, like goods, is infinite.
The only time the market will consume more labor because its cheaper is if the amount of labor they already had was already insufficient, in which case if they really needed more people, they would have already hired them.