Sadly we have this issue but having considerable HR experience we are able to manage it more effectively.
Firstly the PC is the line manager of the Clerk and have dotted line management of the other employees under the line management of the Clerk.
Secondly, if the work is not been carried out quickly enough or to the right standard then the book stops with the Clerk as the line manager. The staffing committee should (with best practice) hold regular one-2-ones with the Clerk including bi-annual or annual performance reviews. Within these reviews, anything should be raised such work not getting done by the Clerk's team and a reason given as to why they are not. The reasons, however, shouldn't simply be accepted time and time again. If the issues are still outstanding or there is a pattern there then the Clerk should be placed on something called a PIP (Personal Improvement Plan). A simple guide on what that is can be found here: PIP Explained
Note: A PIP is not a stage of disciplinary - this is a support tool to encourage the Clerk to do their required tasks to the required standards hence why the objectives should be SMART.
A PIP should be reviewed regularly to ensure the objectives are being met, this is nothing something which should be left for a bi-annual or annual review. If during a PIP review the work is still not being met then the PIP will lead you into a disciplinary stage, which won't necessarily need an investigation as a lot of the work is included in the PIP.
Sadly some Clerk's put themselves into an 'untouchable category' as they feel they are the font of all knowledge and Cllr's have none. For this reason, many issues are not dealt with appropriately because obstacles have been put in front of proper processes. For this reason also don't assume your local association is giving you the best advice, perhaps follow up with ACAS for free.