The statement in CD Part 1 ‘the Employer is not willing to take over the works before the Completion Date’ essentially relates to the Employer not willing to ‘take over’ where the whole of the works is completed EARLIER than the Completion Date.
This statement would particularly relate to the original intended programme of works and I see no reason why the Employer could not ‘change their mind’ on this matter, where circumstances have changed, as it has in your case. It is possible for the Employer to take over part of the works before Completion, as is their right under clause 35.
It sounds like certain parts of the works are being used before Completion. This may be stated in the Works Information, although if that is not the case then the Employer would be required to take over these parts, otherwise the Contractor has no obligation to allow such use before Completion.
Whether taken over or not, the Contractor remains responsible for ‘care of the works’ which may be an issue where the roads and footpaths are currently in use, which could be the reason behind the discussion.