For clarity, I'm proposing an answer to my question, asked a long time ago, but now attracting answers and comments.
It's important to read the question carefully. It is specifically about the summons and the requirement that it be signed by the proper officer. This requirement stems from the Local Government Act 1972 Schedule 12, section 10 (2). Item (a) is easily satisfied, being the requirement for a public notice. Item (b) is the problem, as it stipulates that "a summons to attend the meeting, specifying the business proposed to be transacted at the meeting and signed by the proper officer of the council, shall be left at or sent by post to the usual place of residence of every member of the council" (my bold).
There is some leeway granted in 10(3) where it is said that "Want of service of any such summons as is referred to in sub-paragraph (2)(b) above on any member of the parish council concerned shall not affect the validity of the meeting".
Now the question is, just how far does the leeway extend. Reading it suggests that legislators might have had in mind a situation where a summons is lost in the post or a councillor moves home without telling the clerk, or such like mishaps. Maybe it could extend to the clerk forgetting to post all of the summonses. It's not clear that it is intended to cover the case where the proper officer does not sign the summons.
However, legislators doubtless did not intend to leave parish councils in a situation where they are unable to function should the clerk (as the sole proper officer in most cases) cease to be available for whatever reason and without an alternative proper officer having been appointed. That being the case, it seems to me that the answer to the question is that it is acceptable for the summons to be unsigned in the case where there is no proper officer, relying on a liberal interpretation of 10(3). That allows a valid meeting to be called, at which an interim proper officer can be appointed.
It always was a pedantic point, but it was also a puzzle! An alternative to the liberal interpretation of the legislation would be for the standing orders to be extended to cover the situation and to stipulate that the chairman (or failing chairman, then vice chairman, and failing vice chairman the oldest available councillor) is to become proper officer (unpaid) should the usual proper officer be unavailable, purely for the function of signing the summons. This would anticipate and avoid any problem over the signing of the summons.