I do not believe that the power was given to the Federal Government and I also believe that the "General Welfare Clause" was never meant to include powers that were not reserved for the Federal Government, hence Amendment 10 exists, telling you that such powers are not granted under this clause. Specifically, the "General Welfare" is listed as the reason the Federal Government is granted the specific powers it is granted, it is not a power unto itself and does not grant any extra powers that were not enumerated....
There is a reason we have the 10th, it is for this purpose. So even lawyers can understand that the power of Government is limited to the things that were specified.