I'm not sure we're actually disagreeing on anything FL - and clearly we've both worked extensively with AAIB, albeit in somewhat different roles.
Basically any investigator at that level needs to be professionally qualified, have a healthy mixture of academic training and professional experience.
However, the world - including aviation - is shifting subtly towards a preference for people with higher educational qualifications. For the last 10 years it's become almost impossible to qualify as a Chartered Engineer (and certainly most AAIB engineering inspectors I've worked with hold that qualification) without a masters degree.
Whilst I've not met anybody at AAIB with a PhD either (although they certainly use many external people who do), it is a basic qualification in conducting complex research projects, and somebody who has done a PhD in aerospace engineering certainly has developed useful skills applicable to air accident investigation, even if those skills can also be obtained in other ways.
On the other hand I can't think of anybody I've worked with in the university sector doing aviation safety research who didn't have a PhD.
Basically then - have the right professional qualifications in your field (ATPL, CEng or PhD), and a lot of high quality experience. Anything else is a bonus.