Cultured saltwater pearls do have a bead nucleus inserted along with the bit of donor mantle tissue. This is generally true for cultured saltwater pearls, including Akoya pearls, Tahitians, South Sea pearls, Sea of Cortez pearls, Fijian pearls etc.
Why? Because roundness is such a highly valued quality in cultured pearls. It is the ideal. All other things being equal, a round pearl will sell for more money than a non-round pearl. Certainly non-round pearl shapes are very common among saltwater cultured pearls, but the farmer was trying to get a round pearl.
If an oyster that has had a bead implanted subsequently expels the bead, the resulting pearl is a keshi pearl.