As with almost all issues - it is a decision for 'the council.'
The clerk may compile the agenda (they do not "own" it - the council does), the chair may direct how a meeting progresses through the agenda, but it should be the council as a whole that makes fundamental changes to the construct of established practice.
If there is 'personal history' between the chair and the prospective new county councillor, then they need to grow up and be professional.
The chair should not seek to prevent a ward councillor from communicating higher level info via a PC meeting (if that is established practice) - but the ward Cllr should not seek to progress personal issues via that route either. Of course the current PC chair may not be in that role after the AGM...
Any such fundamental move to change established process should be proposed, voted and carried by a majority of Cllrs and then they can carry the responsibility for closing a legitimate communication channel between county and parish council parishioners.