The fact that the government can tell you to cover your naughty bits is absolutely precedent for the government telling you to cover your mouth and nose. I am pretty sure the feds don't have authority, but state and local governments do.
You are correct.
There is no federal police power akin to the police power of the States and their political subdivisions under the Tenth Amendment.AMENDMENT X [1791]
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
Under the Tenth Amendment (federalism), the States and their political subdivisions have the police power, a sweeping power to enact laws, regulations, and ordinances that promote the health, safety, and welfare of the community .
The States' police power, and that of their political subdivisions, is a comprehensive power that authorizes regulation of everything from A to Z, from the sale of alcoholic beverages to the caging of dangerous animals in zoos.
Commonwealth v. Alger (1851)
The government's power to enact such regulations for the good and welfare of the community as it sees fit, [is] subject to the limitations that the regulation be both reasonable and constitutional.
It is much easier to perceive and realize the existence and sources of this power, then to mark its boundaries, or prescribe limits to its exercise.
JACOBSON v. COMMONWEALTH OF MASSACHUSETTS (1905)
The authority of the state to enact this statute is to be referred to what is commonly called the police power,—a power which the state did not surrender when becoming a member of the Union under the Constitution.
Although this court has refrained from any attempt to define the limits of that power, yet it has distinctly recognized the authority of a state to enact quarantine laws and 'health laws of every description;' indeed, all laws that relate to matters completely within its territory and which do not by their necessary operation affect the people of other states.
According to settled principles, the police power of a state must be held to embrace, at least, such reasonable regulations established directly by legislative enactment as will protect the public health and the public safety.