Sorry Ancient, I too am ancient and don't know.
There doesn't HAVE to be a company appointed as DA, but it is wise. For example, MoD was the DA for some aircraft engine types when we had a dedicated Aero Engine Directorate and, importantly, experts in the field. You do get ridiculous extremes; for example, in 1991 Alcock's regime wanted to cancel all avionic DA contracts and take over DA-ship at Harrogate, led by Supply Managers. However, his incompetence was never in doubt and we headed that off. His reaction was to chop safety funding so we couldn't do the job either, so maybe we would have been better off letting the suppliers (mis)manage it. But it shows you how little very senior people understand about such an important subject.
What the regs say (or said, as they've been cancelled, but I have my own copy here) is that if the original DA does not want the job, is not suitable, or no longer exists, a Design Custodian can be appointed (along with the System Co-ordinating Design Authorities/Custodians, as required). The latter is the basis of all systems integration and functional safety. The MoD must make arrangements for the DC to hold a certified copy of the Master Drawings, and a contractual arrangement must be made with the holder of the Masters for upkeep. A good example is most US-sourced avionics. Joyce Loebl in Gateshead became our best known DC; and a very good job they made of it too. Very often the difficulty is persuading a UK company to take on the responsibility, knowing that in the background there will be hassle with the OEM - especially the US I'm afraid. They treat us like s##t and over-charge to the point of fraud. I remember being quoted a cool £1M (not $) to amend a drawing by inserting a zener diode. (The effect of the diode was to double the available crypto variables in the aircraft, thus meeting Home / Foreign Office requirements following a complaint from Italy).
There are many ways of skinning this cat. MoD could buy the Design Rights and hand them to a UK company, who would become the DA. I don't know if this is what happened on C130K and Marshalls, but it is hard to believe Lockheed sold us the Design Rights. I'd say Marshalls are, more correctly, the Design Custodian. Certainly, during the XV179 inquiry, they didn't have access to data you'd expect a DA to have.
A little known policy, and certainly seldom implemented properly, is that the named representative of the DA or DC is an MoD-approved appointment. If we are considering appointing a company as a DA, we ask them to propose their man, and we (his opposite number in MoD, the named Technical Agency) vet and approve him. We can sack him at any time. (I've only had to do it once, in about 1992). This TA post is precisely the experience the MAA confirm is lacking. Find me someone in DE&S who knows this stuff! (EVERY engineering project manager should, be it Air, Land or Sea).
The reason for this is that, uniquely, he is granted financial and technical delegation powers far in excess of many in MoD. The sole purpose is that he may commit our money and expedite safety matters without seeking MoD approval or contract amendment. This is what makes nonsense of the MoD Commercial claim that only they can commit us to contract. THIS is why the lead question on many investigations should be to this named individual, asking what he did about it when informed. The question is NEVER asked, indicating a breakdown in the safety management system. (The answer would almost always be he hadn't been given the correct delegation). Often, the correct question is asked, but of the wrong company (Chinook ZD576, Red Arrows XX177, Sea King ASaC, Tornado ZG710 for example), which simply illustrates how important is it to establish and record these boundaries of responsibility for all aspects of the programme.