No, the crux of the problem is demand, not supply.
If there were no demand for slaves, there would be no suppliers.
If there was no demand for cocaine, coca would still be nothing more than an obscure plant used by indigenous chewers to mild effect.
Didn’t the Portuguese, French, and Spaniards use their own ships, like the São José, rather than contract all their slave shipping out to Yankees?
They carried them to the Caribbean and to South America. Sugar was big business back then.