The Tenth Amendment, for whatever it's worth, talks about federal and state governments. Local governments: city, town, village, county, etc., are creatures of their state and wholely subordinate.

You may be misinterpreting how “nor” is used in the sentence.

The 10th makes it clear the constitution gives the federal government specific powers which means those specified powers aren’t available to the states or local governments.

The first draft is a little easier to understand: “Each state retains its sovereignty, freedom, and independence, and every power, jurisdiction, and right, which is not by this Confederation expressly delegated to the United States, in Congress assembled.” (Which is not) can be flipped to the front so -> “Those powers given to the United States are its alone, all other powers not specifically listed are retained by the States.”

Both versions makes it clear states are losing some power, but they are given a little spin to make it less obvious.