That is their logic—the problem with it is that people in the immediate area and/or have been in contact with an infected person will be more likely to ask for tests if they know they have been exposed to an infected person.
That helps to reduce the spread of the virus!
That is why most experienced public health officials _want_/_demand_ that the detailed information about the infected person be revealed to the public.
When the secrecy happens, you know that the political officials of the county over-ruled the health professionals of the county.