A councillor can be appointed clerk (or any other council officer) and may continue to be a councillor.
However, a councillor may not receive payment for any council office while they remain a councillor, or for twelve months afterwards.
The clerk is indeed an employee, but these provisions are not to do with general employment legislation (such as minimum wage). They are specific legislation relating to the situation of councillors and council employees. It is obvious that there is a fundamental conflict of interest between being a councillor and also being a paid employee of the council.