The model documents specify that the Clerk will receive a performance review and annual appraisal and that it will be carried out by the Chair (or V-C in Chair's absence), but only on resolution of the Council (see SO 19(c)). So it's an individual councillor, operating under the instruction of the whole council and reporting back to them accordingly.
For practical reasons, as Chloe has highlighted, day-to-day oversight should be the responsibility of an individual and under normal circumstances, that would have to be a councillor, as there simply isn't anybody else. Line management by committee is fraught with challenges. Whether that individual is the Chair or not depends, to some degree, upon the wording of other policies. For example, in a single-employee council, if the Clerk has a grievance, there's a strong possibility that it will relate to actions or omissions of his/her line manager. If the grievance policy states that all grievances will be investigated by the Chair, or that the Chair has a specified role later in the grievance process, it would make sense not to have the Chair as the line manager. The wording of the disciplinary policy has a similar impact.
There's a tendency to think of the line manager as the whip-cracker overseeing the employee with an iron fist. This would fall outside the powers of an individual member of the council, as well as being frowned upon at employment tribunals (as I'm discovering right now!) A line manager should be a supportive friend, a guide through the fog, a shoulder to cry on. It may be wise to choose the line manager based upon their personal qualities, rather than their title. Whoever you choose, they must understand the limit of their powers.
To summarise, there is very little specific legislation on this matter, although basic employment law always applies, so there is leeway to do what's best in differing circumstances. An individual councillor may not make decisions on behalf of the council, but may operate within a framework agreed by the council, so anything specified within a policy document, or any resolution adopted by the council, may be delegated to an individual councillor, as the council has already made the decision.