The county executive elected under s. 59.17 or the county administrator elected or appointed under s. 59.18 shall appoint a county assessor as prescribed in and subject to the limitations of s. 70.99, approve the hiring of the assessor’s staff as prescribed in that section and otherwise comply with that section. In counties with neither a county executive nor a county administrator the appointment of the county assessor shall be the duty of the chairperson of the board subject to the approval of the board and subject to the limitations of s. 70.99. The hiring of the assessor’s staff shall be the duty of the county assessor subject to the limitations of s. 70.99.