They do have the powers, but they have to be able to implement them. It is a symptom of the layers of government in the USA:as I hear it the people are taking things into their own hands and ignoring the governors. something about them not having the powers they think they do to stop people working.
Start at Federal level,
then State level,
followed by county level,
finally city/township level.
Each has its own layer of legislation, rules and law enforcement, and each wants to be able to piss higher up the wall of politics and power than the other. Watching the local and national news here it is quite amazing the drivel that comes out of Washington (read Trumper and Co.) and either the kowtowing to his "xspurt" opinion, or the advocation of commonsense and intent to follow the science by some Mayors of large areas like Atlanta, and Washington DC.
Theoretically State Governors could force mayors to do their bidding, however the language being used by the sensible governors and mayors seems to be very careful. They are advising rather than instructing businesses and individuals how they should conduct themselves. For example: The Mrs had a teleconference with the external legal advisors to their company yesterday and the lawyer stated that any company not sticking 100% to the CDC guidelines could find themselves at the sharp end of a lawsuit in the future should there be death or illness. Should a business follow the direct instruction of a Governor/Mayor and it causes death, or serious illness then they could be vicariously liable. There are also considerations for shared buildings should an employee in one business become ill/die that can impact the other businesses if they fail to perform certain subsequent actions.
Watching it all happen from the safety of my living room sofa and it seems that the motivational driver here is the almighty $. Trump has his own businesses bleeding money at the moment so he himself wants, probably needs, to get punters through his the doors.