From what I've been taught, the body and blood are contained within the Sacred Host which the laypeople receive. The priest takes it under both species by necessity. Since laypeople are only assisting at Mass, it is only the one species that we can receive to get both. Handing out Communion is not actually part of the Mass at all. Only the taking of the Body and Blood during the Priest's communion is part of the Mass.
Both species used to be given out, as I understand it, but there was unnecessary risks, such as illness from drinking out of the same cup, and spilling of that Sacred Blood on the ground. The distribution of the Sacred Host to the people is a lot less risky, as It is under the appearance of bread. If the people were to receive It under the appearance of wine, there was a possibility of It getting spilled on the ground. Great lengths are taken to clean such a very sad mess up. We wouldn't want to tread on It if It did spill on the ground, so the priest had to clean It up very carefully, so that none of His Blood would be trodden on.
If only the priest was taking the Species under the appearance of wine, this would not ever happen, unless by an accident of the priest. It's a lot more likely that someone would spill It than the priest taking It himself.