TO deal with the last point first, my interpretation is that clerks appointed by the governing body do not become school staff in the usual sense of the word.
But the problem really doesn't arise, because you seem to have misunderstood the prohibitions.
There is no legal reason why a member of the school administrative staff should not act as clerk for any aspect of governors' work as long as they are not (in any capacity) members of the governing body.
If such a person, having been appointed to clerking duties, were to become a governor she would have to withdraw fromclerking full meetings and statutory committees, but could still clerk non-statutory committees. So could any other member of the governing body other than the head.
I agree that many schools will have difficulty finding suitable clerks for everything as a result of the new regulations (and the statutory committees, where both skill and discretion are so vital, in particular) but it is not as restrictive as you suggest. The head, perhaps I should add, cannot be clerk in any circumstances.
I would make two points not involving legality. Firstly, remember that it is the governing body that decides whom to appoint as clerk, not the head.
Secondly, the governors will have to be careful that the staff member doing clerking is senior enough and confident enough to be genuinely independent. It is not always easy for an employee to resist the authority of the head if at any time a head is tempted to try to influence him or her.