The council shall designate from among its members a chairperson and vice chairperson who shall serve for one-year terms and who may be reelected. Membership on the council shall not constitute holding a public office, and members of the council shall not be required to take and file oaths of office before serving on the council. A member of the council shall not be disqualified from holding any public office or employment by reason of membership on the council, nor shall one member forfeit the office or employment, by reason of appointment under this chapter, notwithstanding the provisions of any law, ordinance or city charter.