Council shall make all appointments provided for by this Charter by majority vote, to serve for the terms provided in this Charter.
   Council may remove any official which it has appointed; but such removal shall be effected, unless the appointee in writing waives these provisions, only by a vote of five members of Council after a hearing at a regular meeting of Council, following the service upon the appointee of written notice of the hearing and of the reasons for his removal. The appointee shall be entitled to attend such hearing, to be heard himself, to present evidence in his behalf, and to examine witnesses appearing in support of any charge against him. The action of Council in any such matter shall be final, and subject to no appeal.