Actually no, they don't have to be able to manufacture anything. All they need to do is to take on the responsibility for airworthiness issues.
When Apex went up the Swannee in 2008, there was an expectation that the Robin fleet would be 'orphanned'. For a prolonged period, although the TC holder was non-existent and spare parts were difficult to come by, the DGAC, EASA and the CAA refused to consider moving them to Permit or even to Annexe II.
The default position with these NAAs is that aircraft should be on C of A, and it is only in the last resort that a Permit should be used, so they are happy to wait. It might mean that although there is a TC holder, there are no spare parts and the aircraft type could be grounded.
In the case of the Robins, the bankruptcy of Apex meant that all agreements with suppliers fell. After a great deal of lobbying by French clubs, the DGAC and EASA allowed Robins to fly without the normal support, but if there had been a major incident the fleet would have been grounded. It wasn't comfortable when our annual came round that year.