What powers does the state have exclusive of the federal government?
That's not what the 10th amendment preserves. All it states is that anything the federal government opts not to regulate can still be regulated by the States. In other words, it's making clear that that you can't argue that the Constitution, through silence, divested the States of any power.
It's true that the expansive powers granted the federal government by the Commerce Clause and 14th Amendment means there the federal government has few practical limitations on its powers. But that doesn't mean the State is powerless. It's simply subordinate to the federal government (thanks to the Supremacy Clause).