No, that's not what that clause means. It does not allow one state to overturn the laws of all other states. "Priviliges and immunities of citizens" refer to those benefits that are normally understood to apply to citizens by virtue of their being citizens. These include the right to own land, to bring suit before a court, the power of arrest, etc.
I would entertain that notion. But, we have this notion to deal with. The Constitution limits what the Feds can do. Smoe people say that per the 9th and 10th, the States can regulate(meaning they can have the power to regulate) anything no specifically denied to it by the Constitution. The State's powers are spelled out in the State Constitution. A state could make it illegal to wear red shoes. I as a traveler from another state which has no such laws, passess through such a state and am arrested. Does anyone really think that defines a free country? First of all, that would impede "interstate commerse". See, that is actually a rational argument for the Feds trumping any such state laws, which based upon this interpretation of the 9th and 10th, would be fully within the States power to pass such laws.