"... all questions coming or arising before a local authority shall be decided by a majority of the members of the authority present and voting thereon at a meeting of the authority" Local Government Act 1972, Schedule 12, para 39(1). So, no appointment has been made without a vote, which must be minuted as a decision of the council.
Moreover, the chairman must execute a declaration of acceptance of office at the meeting at which the appointment is made, and if this is not done, then the appointment becomes ineffective and the post of chairman vacant. A vice chairman may, but is not required to, execute a declaration of acceptance of office.
A council that does not have an elected chairman is not properly constituted and cannot lawfully act. The first meeting after the post of chairman becomes vacant must appoint a chairman as the first item of business. This is indicated by the legislation you cite.
So it looks as if the whole council is now lacking authority to act as a council until a chairman is appointed and signs the declaration.
It does sound as if the clerk may need to brush up on local council law and procedures.
CORRECTION: The declaration of acceptance must be signed at the meeting if the person is present. In general, it must be signed before the person can officiate as chairman. If present, they must officiate, hence the need to sign at once.
