Sorry, are you suggesting that pointing out an example demonstrating that people are legally distinct and separate from the corporate entities they are associated with is a strawman argument to your premise that, as the Council (a corporate entity) is comprised of Parish Councillors (people), the Council "is" the Parish Councillors? I thought it was the heart of the matter!
Would it also be a strawman argument to point out that if all the councillors resigned, the officers of the Council would remain employed by the Council? Your premise applied here would not only result in leaving all the officers unemployed, but the council ceasing to exist!
If you don't mind me saying, I think your entire premise is flawed as it creates a paradox. Under its conditions, the Parish Council can't exist with no Parish Councillors and you can't be a Parish Councillor if no Parish Council exists. If you look at the actual case, nothing about the being, existence, powers or authority of the Parish Council is derived from the being, existence, powers or authority of a Parish Councillor. A Parish Council is enduring. It's powers, rights, privileges and limitations are delegated to it when it is legally created and through subsequent law passed by government rather than delegated to it from it's Councillors. In fact, any rights a Parish Councillor has are delegated from the Council - as required by law - when a Parish Councillor-elect signs (i.e. accepts) the Council's offer of a position at the table. An "acceptance of office" has legal meaning; you are accepting terms put on you. You aren't directing, ordering or instructing the Council, it is very much the other way round. If you don't submit to the terms the Council puts on you, you don't take your seat. The role is to make decisions for the Council when it asks you to on things it wants a decision on. It's not to "be" the Council.