That doesn't fly - the powers not given to the Feds remain with the state (ie. the people) NOT BACK TO THE UNION. By definition, if I have two items and give you one, I keep the other. By not explicitly giving you the second item, it remains with me. Same with states rights.
The Tenth Amendment doesn't say anything about the Feds. It does mention delegating powers to the United States (the union formed of the individual States) and it mentions powers reserved to the States respectively (the States individually, not as part of the Union.)
The Federal Government and the United States are not the same thing. The United States came to exist before the Federal Government came to exist. The Federal Government is an organizational mechanism of the United States, agreed to by the States Which compose the United States. If the States which compose the United States agreed to amend the Constitution so as to do away with the Federal Government and establish a Constitutional Monarchy as an organizational mechanism, the United States would still exist.