Here is the invitation to the meeting with a bit more information about where the Council is going with the demise of the Interlocal Agreement for the Tacoma Pierce County Board of Health. This item says there will be a new Board of Health and the Department will be under the County Executive.

So the County Executive would control the new Health Department.
From Pierce County:
Remote public participation is provided by calling into 253-215-8782 (Webinar ID: 945-1382-9108
This is notice of Rules and Operations Committee and final Council hearings on Proposal No. 2020-136.
An Ordinance of the Pierce County Council Establishing a County Board of Health and Department of Public Health; Terminating the Agreement Creating a Combined County-City Health Department Known as the Tacoma-Pierce County Health Department Upon a Date Certain; Amending Chapter 2.06 of the Pierce County Code, “Departments and Organizations”; Directing the Clerk to Take Certain Actions; Requesting that the Executive Take Preliminary or Transitional Steps Necessary to Implement this Ordinance Including Forwarding Draft Legislation to Create an Executive Department to Manage Administrative Functions of the Health Board; Amending Pierce County Code References to the Tacoma-Pierce County Health Department Consistent with this Ordinance; and Establishing Effective Dates. (Terminating TPCBH Interlocal)
The Final Council hearing on this proposal is set for December 15, 2020, at 3:00 pm. The final Council hearing will be held in Room 1045, County-City Building, 930 Tacoma Ave S, in Tacoma. Due to the COVID-19 pandemic, the Council Chambers is closed to the public.
The Rules and Operations Committee hearing is set for December 7, 2020, at 10:00 am. Use the following link for the weekly Council Meeting Schedule, which includes the Committee hearing agenda, location and time:
http://www.co.pierce.wa.us/index.aspx?NID=1430
Use the following link for Proposal information, including hearing dates and related documents:
http://online.co.pierce.wa.us/cfapps/council/iview/proposal.cfm?proposal_num=2020-136